Stochastic Loewner Equation and Applications

Summer School, Catalina Island ^{∗} August 28th - September 02nd 2005

Organizers:

Marek Biskup, University of California, Los Angeles John Garnett, University of California, Los Angeles Christoph Thiele, University of California, Los Angeles

∗supported by NSF grant DMS 0400879

Contents

1 Conformal Invariance of Domino Tiling 5 Jason Asher, University of California, Los Angeles . . . . . . . . . . 5 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 More Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Asymptotics of the Coupling Function . . . . . . . . . . . . . 8 1.5 Conformal Invariance of Heights . . . . . . . . . . . . . . . . . 9

2 Intersection exponents for planar Brownian motion 11 Noam Berger, University of California, Los Angeles . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Cascade relations and functional relations . . . . . . . . . . . 13

2.2.1 An easy exponent . . . . . . . . . . . . . . . . . . . . . 13 2.2.2 Extension of ξ and ξ̃ to real argument - part 1 . . . . . 13 2.2.3 π-extremal length and path measure . . . . . . . . . . 13 2.2.4 Cascade relations - part 1 . . . . . . . . . . . . . . . . 15 2.2.5 Extension of ξ and ξ̃ to real argument and Cascade

relations - part 2 . . . . . . . . . . . . . . . . . . . . . 15 2.2.6 Functional relations . . . . . . . . . . . . . . . . . . . . 17

2.3 Locality and exponents for SLE_{6} . . . . . . . . . . . . . . . . 17 2.4 Determining the Brownian intersection exponents . . . . . . . 18

3 Conformal invariance of planar loop-erased random walks 21 Aubrey Clayton, University of California, Berkeley . . . . . . . . . 21 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Hitting probabilities and estimates . . . . . . . . . . . . . . . 22 3.3 Recognizing the driving process . . . . . . . . . . . . . . . . . 24 3.4 Convergence with respect to a stronger topology . . . . . . . . 25

4 The Loewner differential equation and slit mappings 27 Sunhi Choi, Massachusetts Institute of Technology . . . . . . . . . 27 4.1 Introduction and result . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Slit discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 The Loewner equation and conformal welding . . . . . . . . . 30 4.4 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . 31 5 Aggregation in the Plane and Loewner’s Equation 33

Nick Crawford, UCLA . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 An Introduction to Loewner’s Equation . . . . . . . . . . . . . 33 5.2 A Continuous time/space analog of DLA and a Deterministic

Counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 A Method for Estimating the Growth Time . . . . . . . . . . 36

6 Conformal restriction: the chordal case 39 Jesse Goodman, University of British Columbia . . . . . . . . . . . 39 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 The two-sided restriction measures . . . . . . . . . . . . . . . 39 6.3 One-sided restriction . . . . . . . . . . . . . . . . . . . . . . . 41 6.4 SLE_{6} and Brownian motion . . . . . . . . . . . . . . . . . . . 44

7 Basic Properties of SLE 46 Jon Handy, Uiversity of California, Los Angeles . . . . . . . . . . . 46

8 The harmonic explorer and its convergence to SLE_{4} 51 Nam-Gyu Kang, Massachusetts Institute of Technology . . . . . . . 51 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.2 The driving process converges to BM . . . . . . . . . . . . . . 52 8.3 Local Hausdorff convergence to SLE_{4} . . . . . . . . . . . . . . 55 8.4 Convergence in a strong topology . . . . . . . . . . . . . . . . 55

9 Hausdorff dimensions for SLE_{6} 57 Tai Melcher, University of California, Berkeley . . . . . . . . . . . . 57 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9.1.1 Locality and chordal/radial equivalence . . . . . . . . . 58 9.1.2 The construction . . . . . . . . . . . . . . . . . . . . . 59

9.2 Dimension of the trace . . . . . . . . . . . . . . . . . . . . . . 60 9.3 Dimension of the boundary . . . . . . . . . . . . . . . . . . . . 61

10 Basic Properties of SLE 63 Brady Max NewDelman, UCLA . . . . . . . . . . . . . . . . . . . . 63 10.1 The SLE Trace . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

10.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 10.2.2 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 64 10.3 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 65 10.4 Summary of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 67 10.5 Theoem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

11 Critical percolation in the plane 69 Gábor Pete, UC Berkeley . . . . . . . . . . . . . . . . . . . . . . . 69 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 11.2 Elementary percolation bounds . . . . . . . . . . . . . . . . . 70 11.3 The proof of Cardy’s formula . . . . . . . . . . . . . . . . . . 71 11.4 Continuum scaling limit . . . . . . . . . . . . . . . . . . . . . 75

12 Conformal invariance of planar loop-erased random walks and uniform spanning trees Part II — UST 78 Peter Ralph, UC Berkeley . . . . . . . . . . . . . . . . . . . . . . . 78 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 12.3 UST and the Peano curve . . . . . . . . . . . . . . . . . . . . 79 12.4 Convergence of discrete harmonics . . . . . . . . . . . . . . . . 80 12.5 Driving process convergence . . . . . . . . . . . . . . . . . . . 81 12.6 Stronger convergence . . . . . . . . . . . . . . . . . . . . . . . 82

13 Conformal restrictions, highest weights and SLE 84 Shannon Starr, University of California, Los Angeles . . . . . . . . 84 13.1 Background : the Virasoro Algebra . . . . . . . . . . . . . . . 84

13.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . 84 13.1.2 Highest weight representations . . . . . . . . . . . . . . 85 13.1.3 Degenerate representations and det_{n}(c, h). . . . . . . . 87

13.2 Chordal Restriction Measures and Highest Weight Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 13.2.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . 87 13.2.2 Boundary Correlation Functions . . . . . . . . . . . . . 88 13.2.3 Highest Weight Representations of the Witt Algebra . 89 13.2.4 The Markovian Property and Degeneracy . . . . . . . . 90

1 Conformal Invariance of Domino Tiling

after Richard Kenyon [2] A summary written by Jason Asher

Abstract

We present a summary of the results in [2].

1.1 Preliminaries

In the notation of [2], let T denote the tiling of the plane with unit squares centered at the lattice points of ℤ^{2}, colored in a checkerboard pattern such that the origin tile is colored white. We divide the collection of tiles into four equivalence classes based on their cosets modulo 2ℤ^{2}: W_{0} denotes those white tiles with even coordinates, W_{1} denotes those white tiles with odd coordinates, and B_{0} and B_{1} denote those black tiles with odd and even first coordinates, respectively (and hence even and odd respective second coordinates).

A polyomino is a finite collection of tiles from T whose boundary components consist of simple closed lattice paths. As a special case, a domino is a connected polyomino which consists of exactly two tiles. A corner lattice square of a polyomino P is a tile of P adjacent to a corner of the boundary of P, and which lies on the angle bisector of the interior angle (the ’elbow’ of the corner, if you will). We say that a polyomino is even when all of its corner lattice squares are of type B_{1}.

Let P be a connected even polyomino. Remove a black tile d_{0} from P which is adjacent to the outer boundary of P. Then, for each interior boundary lattice path D_{j}, add a black tile d_{j}, called an exposed square, which is adjacent to D_{j}. The resulting object is what is referred to as a Temperleyan polyomino. ”Temperley’s Trick,” a bijection between domino tilings of Temperleyan polyominos and directed essential spanning forests of a certain associated graph, can be used to show that domino tilings exist [1].

To a Temperleyan polyomino P we associate its interior dual graph which has a vertex for each tile in P, and an edge for each adjacent pair of tiles. A domino tiling of P is then precisely a perfect matching of M (recall that a perfect matching of a graph is a collection of edges such that each vertex is the endpoint of precisely one edge in the collection).

Given a domino tiling of P, there is an integer-valued function h defined on the lattice vertices of P, called the height function which is unique up to an additive constant and which determines the given tiling [4]. Indeed, pick a lattice vertex v_{0} of P and define h(v_{0}) = 0. Then, for any other vertex v, choose a lattice path γ which follows only the edges of the dominos and which connects v_{0} to v. Along γ, the values of h change by ±1 as follows: if the edge being traversed has a black tile on its left, then h increases by 1, otherwise h decreases by 1. It is clear that if P is simply-connected (has only one boundary component), then h is well-defined, as the net change in height of a path circling a single domino is 0. More generally, as each boundary component of P has 0 net change in height in a round trip, it is a result of [4] that h is nonetheless still well-defined. Note that h indeed determines the tiling of P as the adjacent vertices with domino edges between them are precisely those that have height difference 1 (the others have height difference 3).

1.2 Main Result

Let U be a bounded, multiply connected region in ℂ with k + 1 smooth boundary components, each with a marked point d_{o}, d_{1}, …, d_{k}. Let P_{ϵ} be a sequence of polyominos with P_{ϵ} ∈ ϵℤ^{2} which approximate U in the following sense: we require that the boundaries of P_{ϵ} are within O(ϵ) of the boundaries of U and we also place some technical restrictions on the local behavior of the boundaries of P_{ϵ} (see sec. 5.3 of [2]). For each ϵ and j, choose a lattice vertex d^{(ϵ_{j})} ∈ P_{ϵ} which is within O(ϵ) of d_{j}. We then have

Theorem 1. The joint distribution of the height variations of the d_{j}^{(ϵ)} with respect to the uniform measure on domino tilings of P_{ϵ} tends to a finite limit which is conformally invariant.

(By ‘height variations’ we mean the differences of the heights from their mean values.)

1.3 More Preliminaries

First, we define a few operators on ℂ^{ℤ^{2}}. For a function f : ℤ^{2} → ℂ, let ∂_{x} and ∂_{y} be given by

∂_{x}f(v) = f(v + 1) − f(v − 1), ∂_{y}f(v) = f(v + i) − f(v − i).

Then, let ∂_{z} = ∂_{x} − i∂_{y} and ∂z̄ = ∂_{x} + i∂_{y}. We say a function F ∈ ℂ^{B} (i.e. a function which vanishes on white vertices) is discrete analytic if it is real on B_{0}, purely imaginary on B_{1}, and if ∂z̄F = 0. If we write F = f + ig for f ∈ ℝ^{B_{0}} and g ∈ ℝ^{B_{1}}, then a discrete analytic function is precisely one which satisfies the discrete Cauchy-Riemann equations

∂_{x}f(v) = ∂_{y}g(v) for v ∈ W_{0}  (1)
∂_{y}f(v) = −∂_{x}g(v) for v ∈ W_{1}.  (2)

If F satisfies these equations at all but finitely many vertices v_{i} ∈ W, we say that F is discrete analytic with poles at these v_{i}. Finally, note that −∂_{z}∂z̄ is (4 times) the usual Laplacian on B_{0}, i.e. for f ∈ ℝ^{B_{0}} we have

−∂_{z}∂z̄f(v) = 4f(v) − f(v + 2) − f(v + 2i) − f(v − 2) − f(v − 2i).

A similar equation holds for g ∈ ℝ^{B_{0}}, and so we have that the real and imaginary parts of a discrete analytic function are harmonic: ∂_{z}∂z̄(f + ig) = ∂_{z}(0) = 0, and thus Δf = 0 and Δg = 0 for the Laplacian on B_{0} and B_{1} respectively.

For a subgraph M of ℤ^{2}, let K be the adjacency matrix of M with weighted edges so that at each white vertex the counter-clockwise weights are 1, i, −1, −i, respectively, starting at 3-o’clock. K is the Kesteleyn matrix of M, and Kestelyn showed that for a finite connected graph, the absolute value of the determinant of K is the square of the number of perfect matchings of M. A simple calculation shows that if M = ℤ^{2}, as an operator on ℂ^{B} , K is simply the operator ∂z̄, and K^{∗}K is the Laplacian on both B_{0} and B_{1}.

More specifically, let P be a Temperleyan polyomino, and let M be its interior dual graph. Let E be a finite collection of disjoint edges of M, and b_{1}, …, b_{k} and w_{1}, …, w_{k} be the black and respectively white vertices covered by E. Then, if μ is the uniform probability measure on the perfect matchings of M, we have the following result of Kenyon:

Theorem 2. [3] The μ-probability that E occurs in a perfect matching is given by |det(K_{E}^{−1})|, where K_{E}^{−1}is the submatrix of K^{−1} whose rows are indexed by b_{1}, …, b_{k} and whose columns are indexed by w_{1}, …, w_{k}.

(Note: This is Theorem 7 in [2])

The matrix K^{−1}, when considered as a function from M×M to ℂ, is called the coupling function, and is denoted by C. Note that KC(v_{1}, v_{2}) = δ_{v_{1}}(v_{2}).

1.4 Asymptotics of the Coupling Function

One of the central results of [2] is that the scaled discrete analytic function (1/ϵ)C(v_{1}, ·) converges to a pair of complex-analytic functions F_{0}, F_{1} as ϵ tends to 0. (When v_{1} ∈ W_{j}, the limiting function is F_{j} for j = 0, 1). Furthermore, these limiting functions transform analytically under conformal mappings. We will give a description of the functions F_{j}, but their existence and uniqueness from the given definitions will be part of the proof of the below. In all of the following, U is a region in ℂ with smooth boundary components D_{0}, D_{1}, … D_{k}, labeled so that D_{0} is the outer component, and each with a marked point d′_{j}. Also, z_{i} are points in U for i = 1, 2 with z_{1} interior.

For k = 0, 1, F_{k}(z_{1}, z_{2}) is analytic as a function of z_{2} for a fixed z_{1} but with a pole at z_{2} = z_{1} of residue 1/π. Also, F_{k} has no other poles on \overline{U}, with the possible exception of the d′_{j} for j ≠ 0. Finally, F_{0} and F_{1} are both zero at d′_{0}, however on ∂U, F_{0} has zero real part, while F_{1} has zero imaginary part.

For each sufficiently small ϵ > 0, let P_{ϵ} be a Temperleyan polyomino in ϵℤ^{2} which approximates U as in the preamble to Theorem 1. Let M_{ϵ} be the interior dual graph of P_{ϵ}, and let v_{1} and v_{2} be a white, and black respectively, vertex of M_{ϵ}. Then we have

Theorem 3. [2] Fix any real ξ > 0. The coupling function C(v_{1}, v_{2}) on the graph M_{ϵ} satisfies: for v_{1} ∈ W_{0} and v_{1}, v_{2} not within ξ of the boundary of M_{ϵ},

\frac{1}{ϵ}C(v_{1}, v_{2}) = \frac{1}{ϵ}C_{0}(v_{1}, v_{2}) + F_{0}^{∗}(v_{1}, v_{2}) + o(1),

where F_{0}^{∗} is defined by the condition that F_{0}(z_{1}, z_{2}) = 1/(π(z_{2} − z_{1})) + F_{0}^{∗}(z_{1}, z_{2}), and where C_{0} is the coupling function on ϵℤ^{2}.

If v_{1} ∈ W_{1}, rather, then,

\frac{1}{ϵ}C(v_{1}, v_{2}) = \frac{1}{ϵ}C_{0}(v_{1}, v_{2}) + F_{1}^{∗}(v_{1}, v_{2}) + o(1),

where F_{1}^{∗} is defined by the condition that F_{1}(z_{1}, z_{2}) = 1/(π(z_{2} − z_{1})) + F_{1}^{∗}(z_{1}, z_{2}).

(Note: This is Theorem 13 in [2])

In the above theorem, the equalities need to be appropriately interpreted: For the first, this means that for v_{1} ∈ W_{0} and v_{2} ∈ B_{0}, the left-hand side equals the real part of the right hand side, and that for v_{1} ∈ W_{0} and v_{2} ∈ B_{1}, the left-hand side equals i times the imaginary part of the right hand side. For the second, this means that for v_{1} ∈ W_{0} and v_{2} ∈ B_{1}, the left-hand side equals the real part of the right hand side, and that for v_{1} ∈ W_{0} and v_{2} ∈ B_{0}, the left-hand side equals i times the imaginary part of the right hand side. Furthermore, when v_{1} and v_{2} are far apart (i.e. not within o(1)), by a lemma one has the following simplified form:

\frac{1}{ϵ}C(v_{1}, v_{2}) = F_{j}(v_{1}, v_{2}) + o(1), for j = 0, 1 where v_{1} ∈ W_{j}.

We have a similar result for the case when v_{1} is close to a horizontal flat boundary of P_{ϵ}, and which in the simplified form for v_{1}, v_{2} far apart gives the same equation as above.

Finally, let F_{+} = F_{0} +F_{1} and F_{−} = F_{0} −F_{1}, and note how these functions transform under conformal mappings:

Proposition 4. [2] The function F_{+}(z_{1}, z_{2}) is analytic as a function of both variables. The function F_{−}(z_{1}, z_{2}) is analytic as a function of z_{2} and antianalytic as a function of z_{1}. If V is another domain with smooth boundary and if f : U → V is a bijective complex analytic map sending the marked points on U to those of V, then

F_{+}^{U}(v, w) = f′(v)F_{+}^{V} (f(v), f(w)),
F_{+}^{U}(v, w) = \overline{f′(v)}F_{+}^{V} (f(v), f(w)).

(Note: This is Proposition 15 in [2].)

1.5 Conformal Invariance of Heights

Recall the notation from the previous section, and for each j let e′_{j} ≠ d′_{j} be another point of D_{j}. Choose a vertex e_{j} on the boundary of P_{ϵ} which is within O(ϵ) of e′_{j}. Finally, let P_{ϵ} be a family of Temperleyan polyominos which approximate U as above, with the additional requirement that in a neighborhood of each e′_{j} the boundary of P_{ϵ} is flat and horizontal with the interior of U locally below e_{j}. Let h_{j} be a random variable which gives the height of e_{j} for a random tiling, assuming that the height of e_{0} is 0, and let h̄_{j} be the mean value of h_{j}.

The key to Theorem 1 is the computation of the moment

E((h_{1} − h̄_{1})^{n_{1}}(h_{2} − h̄_{2})^{n_{2}} ⋯ (h_{k} − h̄_{k})^{n_{k}})

for positive integers n_{j}. Letting K = n_{1} + ⋯ + n_{k}, this is done by the following:

Proposition 5. Let {γ_{j}}_{j=1}^{K} be a collection of pairwise disjoint paths in U such that for each j there are n_{j} paths running from the outer boundary to the j^{th} boundary component. Then, as ϵ tends to zero, the above moment converges to

(−i)^{k} ∑_{ϵ_{1},...,ϵ_{k}∈[±1]} ϵ_{1} ⋯ ϵ_{k} ∫_{γ_{1}} ⋯ ∫_{γ_{k}} det(F_{ϵ_{i},ϵ_{j}}(z_{i}, z_{j}))dz_{1}^{(ϵ_{1})} … dz_{1}^{(ϵ_{1})}

where dz_{j}^{(−1)} = \overline{dz_{j}}, and F_{ϵ_{i},ϵ_{j}} = F_{sgn(ϵ_{i})}^{(ϵ_{j})} for i ≠ j and 0 otherwise.

(Note: This is Proposition 22 in [2].)

By the results from the previous section, each integrand in this moment is conformally invariant, and hence so is the moment. The rest of the proof consists of verifying that the corresponding moment generating function has a nonzero radius of convergence about 0, from which it follows that there is a unique underlying limiting distribution and that it is the limit of the underlying distributions for the nonzero ϵ.

References

[1] Burton, R. and Pemantle, R. Local characteristics, entropy and limit theorems for spanning treet and domino tilings via transfer-impedances Annal. Probab. 21 (1993), 1329-1371

[2] Kenyon, R., Conformal invariance of domino tiling. Annal. Probab. 28 no.2 (2000), 759-795

[3] Kenyon, R., Local statistics of lattice of lattice dimers Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 591-618

[4] Thurston, W. P., Conway’s tiling groups. Amer. Math. Monthly 97 (1990) 757-773

Jason Asher, University of California, Los Angeles email: asherj@math.ucla.edu

2 Intersection exponents for planar Brownian motion

after G. Lawler, O. Schramm and W. Werner A summary written by Noam Berger

Abstract

We define the intersection exponents and give a (very) rough sketch of how they are calculated.

2.1 Introduction

Suppose that n Brownian motions, β^{1}, …, β^{n} are started at n distinct points on the upper half plane, and we let them run up to time t. We consider the probability of the event that no two paths intersect, i.e.

f_{n}(t) = P [∀_{i≠j}(β^{i}[0, t] ∩ β^{j}[0, t] = ∅)] .  (1)

In addition, we are interested in the ”half-plane intersection probability”, i.e. the probability that the path do not intersect, and, in addition all paths stay in the upper-half plane:

f̃_{n}(t) = P [∀_{i≠j}(β^{i}[0, t] ∩ β^{j}[0, t] = ∅), ∀_{i}β^{i}[0, t] ⊆ H] .

Clearly, f_{n} and f̃_{n} depend on the initial points. However, their asymptotic behavior does not. The following proposition follows from subadditivity and from estimates of the exit time of a disc of certain radius:

Proposition 1. There exist constants ξ_{n}, n = 1, 2, … and ξ̃_{n}, n = 1, 2, … such that

f_{n}(t) = t^{\frac{1}{2} ξ_{n}+o(1)} and f̃_{n}(t) = t^{\frac{1}{2} ξ̃_{n}+o(1)}.  (2)

The constants ξ_{n}, n = 1, 2, … are called plane intersection exponents, and the constants ξ̃_{n}, n = 1, 2, … are called half-plane intersection exponents. The choice of the constant \frac{1}{2} in (2) comes from the fact that by time t the Brownian motion is typically at distance t^{\frac{1}{2}} from its initial point. Therefore, the probability of non-intersection of the paths up to the first time they reach distance r from the origin is r^{ξ_{n}+o(1)} and r^{ξ̃_{n}+o(1)} for the half-plane intersection. With a slight abuse of notation, we will use f_{n}(r) and f̃_{n}(r) for the probabilities that the Brownian motions hadn’t intersected (or left H for f̃) before reaching distance r from the origin.

The main result, proven in [1, 2] and based on results of [4] is the following:

Theorem 2. For every n,

ξ_{n} = \frac{4n^{2} − 1}{12}  (3)

and

ξ̃_{n} = \frac{n(2n + 1)}{3}.  (4)

Packets of Brownian motions: Let p_{1}, …, p_{n} be positive integer numbers, and let z^{(1)}, …, z^{(n)} be distinct points in the half plane. For every 1 ≤ k ≤ n, let β_{j}^{k}, j = 1, …, p_{k} be p_{k} independent Brownian motions starting at z^{(k)}. Let

B_{k}(t) = ⋃_{j=1}^{p_{k}} β_{j}^{k}[0, t].

Then equivalently to (1) we define

f_{p_{1},...,p_{n}}(t) = P [∀_{i≠j}(B_{i}(t) ∩ B_{j}(t) = ∅)] .  (5)

and

f̃_{p_{1},...,p_{n}}(t) = P [∀_{i≠j}(B_{i}(t) ∩ B_{j}(t) = ∅) and ∀_{i=1,...n}B_{i} ⊆ H] .  (6)

Equivalently, we define the exponents ξ_{p_{1},...,p_{n}} and ξ̃_{p_{1},...,p_{n}} to be so that

f_{p_{1},...,p_{n}}(t) = t^{\frac{1}{2} ξ_{p_{1},...,pn}+o(1)} and f̃_{p_{1},...,p_{n}}(t) = t^{\frac{1}{2} ξ̃_{p_{1},...,pn}+o(1)}.  (7)

The following theorem is a generalization of Theorem 2.

Theorem 3.

ξ̃_{p_{1},...,p_{n}} = \frac{[\sqrt{24p_{1} + 1} + … + \sqrt{24p_{n} + 1} − (n − 1)]^{2} − 1}{24},  (8)

and

xi_{p_{1},...,p_{n}} = \frac{(\sqrt{24p_{1} + 1} + … + \sqrt{24p_{n} + 1} − n)^{2} − 1}{48}.  (9)

Theorem 2 follows immediately from Theorem 3. Theorem 3 is proved in [3], whereas [1] and [2] prove (8) and (9) only for a subset of all possible vectors, a subset that nevertheless contains the vectors (1, …, 1) of every length, thus proving Theorem 2. The rest of this summary will be dedicated to sketching the main ingredients of the proof of Theorem 2 the way it was done in [4, 1, 2].

2.2 Cascade relations and functional relations

2.2.1 An easy exponent

The following Lemma is useful and easy.

Lemma 4. ξ̃_{1} = 1.

2.2.2 Extension of ξ and ξ̃ to real argument - part 1

Let p and λ be positive integers. Then, it is easy to see that

f_{p,λ}(t) = E [P (β_{1}^{1}[0, t] ∩ B_{2}(t) = ∅)^{λ}| B_{2}(t)] .  (10)

(10) gives a natural extension of f and therefore of ξ to the case when λ is a positive real number. This way one can extend

ξ_{λ_{1},p_{1},λ_{2},...,p_{n},}

and

ξ̃_{λ_{1},p_{1},λ_{2},...,p_{n},λ_{n+1}}

to the case where the λ_{k}-s are real rather than integer. As we shall see, the generalized version of the exponents plays a crucial role in calculating the values of the exponents.

2.2.3 π-extremal length and path measure

Since the exponents ξ and ξ̃ do not depend on the initial points, we may start the Brownian motions at any arbitrary point, or at a random point. A natural choice is to start the Brownian motions at independent uniform points on the circle ∂U = {z : |z| = 1} for the whole-plane exponents and the half circle ∂U ∩ H for the half plane exponents.

We now turn our attention to the half plane. The logarithm send H conformally into the strip {z : 0 < ℑz < π}. Therefore f̃_{n}(r) could be the probability that n Brownian motions starting in different points along the line {z : Re(z) = 0 ; 0 < ℑz < π} do not intersect and do not leave the strip before their arrival at {z : Re(z) = log r ; 0 < ℑz < π}. It turns out that if we condition on the event that real parts of the Brownian motions remain positive, we do not change the value of the exponents. (note that some caution is needed here b/c the event at hand is of probability zero).

Therefore, we work inside a domain (rectangle) whose boundary is divided into 4 arcs, and estimate the quantity of ensembles of paths starting from one arc, reaching the opposite arc, and in addition not intersecting each other and not hitting the perpendicular arcs. Path measure: Therefore, it seems natural to define a measure on paths in a simply connected domain, starting and ending at the boundary, that will be conformally invariant and would extend the notion of conditioned Brownian motion in a rectangle. Such a measure exists, and is denoted by μ. μ is an infinite measure, but every event that requires some lower bound on the distance between start and end points of a path is of finite measure. In particular, on the rectangle with arcs

A = [0, iπ] ; B = [iπ, log r + iπ]
C = [log r + iπ, log r] ; D = [log r, 0],

we get

μ × … × μ (β_{1}, …, β_{n} start at A and end at C and no two intersect) = r^{ξ̃_{n}+o(1)}

π-extremal length: For every simply connected Jordan domain D with four distinguished points a, b, c, d on its boundary, there exists a unique L such that there exists a conformal mapping f from D to R_{L} = {z : 0 < ℜz < L ; 0 < ℑz < π} with

f(a) = 0 ; f(b) = iπ ; f(c) = log L + iπ ; f(d) = log L.

L is called the π-extremal length of (D, a, b, c, d) and denoted L(D, a, b, c, d). For ease of notation, if D is not connected or is empty then L(D) = ∞. The domains that usually interest us are parts of rectangles below and above Brownian paths and SLE_{6} hulls.

2.2.4 Cascade relations - part 1

Let p_{1}, …, p_{n} be positive integers. Let B_{1}, …, B_{n} be packets of, respectively, p_{k} Brownain motions starting at uniform points on [0, iπ]. Let D_{1} be the part of the rectangle of length log r above B_{1}. For k = 2, …, n let D_{k} be the part of the rectangle below B_{k−1} and above B_{k} and let D_{n+1} be the part of the rectangle below B_{n}. Note that D_{k} may be empty or disconnected.

In view of Lemma 4 we now get the following:

Proposition 5. Let p_{1}, …, p_{n} be positive integers and let λ_{1}, …, λ_{n+1} be positive reals. Then,

E [E (e^{−λ_{1}L(D_{1})−...−λ_{n+1}L(D_{n+1})})| B_{1}, …, B_{n}] = r^{−ξ̃_{λ_{1},p_{1},...,pn,λ_{n+1}}+o(1)}

From Proposition 5 and, again, using Lemma 4, we get the following relations, known as cascade relations.

Proposition 6. 1. Let p_{1}, …, p_{n} be positive integers and let λ_{1}, …, λ_{n+1} be positive reals. Then,

ξ̃ (λ_{1}, p_{1}, …, p_{n−1}, λ_{n}, p_{n}, λ_{n+1})
= ξ̃ (λ_{1}, p_{1}, …, p_{n−1}, ξ̃(λ_{n}, p_{n}, λ_{n+1})) .  (11)

2. Let p_{1}, …, p_{n} be positive integers and let λ_{1}, …, λ_{n} be positive reals. Then,

ξ (p_{1}, λ_{1}, …, p_{n−1}, λ_{n−1}, p_{n}, λ_{n})
= ξ (p_{1}, λ_{1}, …, p_{n−1}, ξ̃(λ_{n−1}, p_{n}, λ_{n})) .  (12)

Note that for notational convenience we put the arguments of ξ̃ and ξ in parentheses rather than in subscript.

2.2.5 Extension of ξ and ξ̃ to real argument and Cascade relations - part 2

Iterating (11) and (12) we get that for all k < n,

ξ̃ (λ_{1}, p_{1}, …, p_{n−1}, λ_{n}, p_{n}, λ_{n+1})
= ξ̃ (λ_{1}, p_{1}, …, p_{k}, ξ̃(λ_{k+1}, p_{k} + 1, …, p_{n}, λ_{n+1})) .

and

ξ (p_{1}, λ_{1}, …, p_{n−1}, λ_{n−1}, p_{n}, λ_{n})
= ξ (p_{1}, λ_{1}, …, p_{k}, ξ̃(λ_{k}, …, p_{n}, λ_{n})) .

Combining this with the fact that ξ̃ is invariant under reflection, we get that ξ and ξ̃ are invariant under every permutation of the p-s and under every permutation of the λ-s.

We now define ξ̃(α_{1}, …, α_{n}) for all positive real numbers α_{1}, …, α_{n}. The function κ̃(λ) = ξ̃(1, λ) is bijective from [0, ∞) to [1, ∞). Therefore, remembering that ξ̃(λ) = λ we can define

ξ̃(α, α′) = κ̃^{−1}(ξ̃(α, 1, α′))  (13)

and then, inductively

ξ̃(α_{1}, …, α_{n}) = ξ̃(α_{1}, ξ̃(α_{2}, …, α_{n})).  (14)

We similarly extend ξ to general argument α_{1}, …, α_{n}, but then we need to restrict ourselves to the case where at least two of the α_{k}-s are at least 1: Assuming α_{1} ≥ 1 we take β_{1} to be so that α_{1} = ξ̃(1, β_{1}) and then define

ξ(α_{1}, …, α_{n}) = ξ(1, ξ̃(β_{1}, α_{2}, …, α_{n})).  (15)

The requirement that there exists another α_{i} which is at least 1 is needed in order to guarantee that (15) is, indeed, a proper extension of ξ.

Theorem 7. The extensions of ξ̃ and ξ defined in (13), (14) and (15) are the unique extensions such that

1. ξ and ξ̃ are invariant under permutations.

2. ξ and ξ̃ satisfies

ξ̃(α_{1}, …, α_{n}) = ξ̃(α_{1}, …, α_{k}, ξ̃(α_{k+1}, α_{n}))  (16)
ξ(α_{1}, …, α_{n}) = ξ(α_{1}, …, α_{k}, ξ̃(α_{k+1}, α_{n}))  (17)

for every k < n.

The relations (16) and (17) are called the generalized cascade relation.

2.2.6 Functional relations

For α and n we use the notation α^{⊗n} for the n-fold repetition of α.

Proposition 8. The limit

lim_{n→∞} \frac{1}{n^{2}} ξ̃(α^{⊗n})

exists for every α, and is continuous and strictly increasing.

We define

U^{2}(α) := lim_{n→∞} \frac{1}{n^{2}} ξ̃(α^{⊗n})  (18)

From the cascade relations and Proposition 8, we learn the following important fact.

Theorem 9. For every α_{1}, …, α_{n},

U(ξ̃(α_{1}, …, α_{n})) = U(α_{1}) + … + U(α_{n}).  (19)

In particular U contains all of the information about ξ̃.

From Theorem 7 we can deduce the following fact:

Theorem 10. There exists a continuous and strictly increasing function η such that

ξ(α_{1}, …, α_{n}) = η (ξ̃(α_{1}, …, α_{n}))  (20)

for all n-tuples α_{1}, …, α_{n} such that at least two of the α-s are at least 1.

2.3 Locality and exponents for SLE_{6}

The following two theorems are the heart of the calculation of the intersection exponents.

Theorem 11 (Locality). Let D and D^{∗} ⊆ D be Jordan domains and let a be a boundary point of both D and D^{∗}. Let K_{t} be the hull of SLE_{6} in D starting at a towards a point on ∂D − ∂D^{∗} and let Let K_{t}^{∗} be the hull of SLE_{6} in D^{∗} starting at a towards a point on D ∩∂D^{∗}. Let τ = sup{t : K_{t} ∪D ∩∂D^{∗} = ∅} and τ^{∗} = sup{t : K_{t}^{∗} ∪ D ∩ ∂D^{∗} = ∅}. Then K_{t} up to τ and K_{t}^{∗} up to τ^{∗} have the same distribution up to a (random) time change.

Theorem 11 enables us to obtain cascade-type relations between SLE_{6} exponents and Brownian motion exponents. Then, once the exponents for SLE_{6} are known, one can use these relations to calculate the exponents for Brownian motion. The Theorem below tells us what the SLE_{6} exponents are.

Let R_{L} be the rectangle of length L and width π. Let S be the hull of chordal SLE_{6} in R_{L} starting at iπ and stopped when it hits [0, L]∩[L, L+iπ]. Let L be the π-extremal length of the region below S, between the arcs [0, iπ] and [L, L + iπ]. If S ∩ [0, L] ≠ ∅ we take L = ∞.

Theorem 12. For every λ ≥ 0, let

u(λ) = \frac{6λ + 1 + \sqrt{24λ + 1}}{6}.  (21)

Then, as L → ∞,

E[exp(−λL)] = exp (−u(λ)L + O(1)(λ + 1)) .  (22)

2.4 Determining the Brownian intersection exponents

In this final section we will see how the half-plane exponents are determined. The whole plane exponents are determined using similar ideas.

The heart of the calculation is in the following lemma.

Lemma 13. For every λ > 0,

ξ̃(1/3, λ) = u(λ).  (23)

where u(λ) is as in (21).

(3) of Theorem 2 follows from Lemma 13 and the cascade relations.

Proof of Lemma 13 (copied almost verbatim from [1]). Let S be as before and let B be a Brownian excursion in R_{L} sampled according to μ. Let x_{s} and x_{e} the start and end points of B. Consider the events

E_{1} = {x_{s} ∈ [0, π] and x_{e} ∈ [L, L + π]}
E_{2} = {S ∩ [0, L] = ∅}
E_{3} = E_{1} ∩ E_{2} ∩ {S ∩ B = ∅}

Let L_{S} be the π-extremal length between the vertical edges of R_{L} in R_{L} −S. Let R_{B}^{+} be the component of R_{L} − B above B and let R_{B}^{−} be the component of R_{L} − B below B Let L_{b}^{−} (resp. L_{b}^{−}) be the π-extremal length of R_{B}^{−} (resp. R_{B}^{−}). Let L_{SB} denote the π-extremal length of the region between S and B.

Let λ > 0. We are interested in the asymptotic behavior of

f(L) = E[1_{E_{3}} exp(−λL_{SB})]

when L = log r → ∞. By first taking expectations with respect to B , and using Theorem 11 for the domains R and R_{B}^{+}, it follows that as L → ∞,

f(L) = E_{B}[E_{S}[exp(−λL_{SB})]]
= E_{B}[exp(−u(λ)L_{B}^{+} + O(1))]
= exp(−ξ̃(1, u(λ))L + o(L)) .

On the other hand, we may first take expectation with respect to S. Given S, the law of L_{SB} is the same as that of L_{B}^{−}, by complete conformal invariance of the excursion measure. Hence, as L → ∞,

f(L) = E_{S} [E_{B} [1_{E_{3}} exp(−λL_{SB})]]
= E_{S} [E_{B} [1_{E_{3}} exp(−λL_{B}^{−})]]
= E_{B} [E_{S} [1_{E_{3}} exp(−λL_{B}^{−})]]
= E_{B} [P_{S}[E_{3} | L_{B}^{+}] exp(−λL_{B}^{−})]
= E_{B} [exp(−L_{B}^{+}/3 + O(1)) exp(−λL_{B}^{−})]
= exp(−ξ̃(1/3, 1, λ)L + o(L))
= exp(−ξ̃(1, ξ̃(1/3, λ))L + o(L)) ,

by the cascade relations (16). Comparing with (24) gives ξ̃(1, ξ̃(1/3, λ)) = ξ̃(1, u(λ)). Finally,

ξ̃(1/3, λ) = u(λ) ,

follows, since λ′ ↦ ξ̃(1, λ′) is strictly increasing.

References

[1] G. F. Lawler, O. Schramm and W. Werner (2001) Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187, no. 2, 237–273.

[2] G. F. Lawler, O. Schramm, W. Werner (2001) Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187, no. 2, 275– 308.

[3] G. F. Lawler, O. Schramm, W. Werner (2002) Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincar Probab. Statist. 38, no. 1, 109–123.

[4] G. F. Lawler, W. Werner (1999) Intersection exponents for planar Brownian motion. Ann. Probab. 27, 1601-1642

3 Conformal invariance of planar loop-erased random walks

after G. Lawler, O. Schramm, and W. Werner A summary written by Aubrey Clayton

Abstract We present the authors’ proof that loop-erased random walk in a domain D converges weakly to radial SLE_{2} in D as the step-size converges to 0. It follows that the scaling limit is conformally invariant.

3.1 Introduction

Let D ⊂ ℂ be a simply connected domain with 0 ∈ D and D ≠ ℂ. For δ > 0, let (Γ(n), 0 ≤ n ≤ T) be a simple random walk on the lattice δℤ^{2}, started at 0 and stopped upon exiting D. The loop-erased random walk (LERW) on D is defined inductively: β_{0} = 0, β_{n+1} = Γ(k), where k = 1 + max{m ≤ T : Γ(m) = β_{n}}. Our goal is to analyze the behavior of this random path in the limit as δ → 0. Since simple random walk converges to Brownian motion in D, we would expect this scaling limit to be something like a ”loop-erased Brownian motion.” However, since Brownian motion contains loops at every scale (and no ”first” loop), the naive approach can not succeed, and we are led to describe the scaling limit in other terms, namely SLE.

Let 𝕌 = {|z| < 1} be the unit disk in ℂ. Given any continuous function W : [0, ∞) → ∂𝕌 (called the driving function), let g_{t}(z) be the solution of Loewner’s (radial) differential equation:

∂_{t}g_{t}(z) = −g_{t}(z)\frac{g_{t}(z) + W (t)}{g_{t}(z) − W (t)}

with initial condition g_{0}(z) = z for all z ∈ 𝕌. This solution exists up to some time τ(z), satisfying lim_{t↑τ(z)}(g_{t}(z) − W (t)) = 0, and the sets K_{t} := {z ∈ 𝕌 :

τ(z) ≤ t} form an increasing family which under some assumptions on W can be shown to be a simple path. In particular, if W (t) = exp(iB_{κt}), where B_{t} is a Brownian motion, the resulting family K_{t} is known to be a simple curve for κ ≤ 4, and this random path is called (radial) SLE_{κ}.

By radial SLE_{κ} in the domain D, we will mean the image of radial SLE_{κ} in 𝕌 under a conformal map from 𝕌 to D fixing 0.

Finally, on the space of unparametrized paths in ℂ, we define the metric ρ(β, γ) := inf sup{|β̂(t) − γ̂(t)| : t ∈ [0, ∞)}, where the infimum is taken over all possible parametrizations β̂ and γ̂ in [0, ∞) of β and γ. The main result of the paper is the following:

Theorem 1. With respect to the metric ρ, the law of LERW on δℤ^{2} ∩ D converges weakly to the law of radial SLE_{2} in D as δ → 0.

That is, for all ϵ > 0, P [{ρ(γ_{δ}, ϕ ∘ β) < ϵ}] → 1 as δ → 0, where γ_{δ} is the LERW on δℤ^{2} ∩ D, β is the radial SLE_{2} path in 𝕌, and ϕ : 𝕌 → D is a conformal map satisfying ϕ(0) = 0 and ϕ′(0) > 0. Since SLE is conformally invariant (in the sense that given another simply connected domain D′ ≠ ℂ and a conformal map f : D → D′, the distribution on paths of SLE in D′ is given by f composed with the distribution on paths of SLE in D), it follows that the scaling limit of LERW is conformally invariant.

The proof will consist of two steps: first, in sections 2 and 3, we will show the convergence of LERW to SLE_{2} with respect to a weaker topology, namely, the topology of convergence of the driving processes, treating LERW itself as a Loewner evolution. Then, in section 4, some regularity properties of LERW will imply the convergence with respect to the stronger topology.

3.2 Hitting probabilities and estimates

Instead of considering the random walk on δℤ^{2} ∩ D for a fixed D and letting δ → 0, we will fix the lattice ℤ^{2} and consider the simple random walk Γ on ℤ^{2} ∩ D as the size of D → ∞. In particular, we will interested in the the quantity rad_{0}(D) := inf{|z| : z ∉ D}. Consider the hitting probabilities H(a, b), defined as the probability that a simple random walk started from a and stopped upon exiting D will hit b. (We will require a to be in V (D) := ℤ^{2} ∩ D, but b may belong to the set of boundary vertices V_{∂}(D), i.e., those lattice points adjacent to a point in V (D).)

Given any such D, we define ψ = ψ_{D} : D → 𝕌 to be the unique conformal map from D onto 𝕌 such that ψ_{D}(0) = 0 and ψ′(0) > 0. Letting

λ(w, u : D) = \frac{1 − |ψ(w)|^{2}}{|ψ(w) − ψ(u)|^{2}} = ℜ (\frac{ψ(u) + ψ(w)}{ψ(u) − ψ(w)})

we have that λ is the density at u of harmonic measure on ∂D started at w with respect to harmonic measure started at 0, and so it is a continuous analogue of the function h(w) := H(w, u)/H(0, u). It is therefore not surprising that the following holds:

Theorem 2. For all ϵ > 0, there exists r_{0} > 0 such that for any domain D with rad_{0}(D) > r_{0} and any points w ∈ V (D) and u ∈ V_{∂}(D) with |ψ(w)| ≤ 1 − ϵ and H(0, u) ≠ 0, we have |\frac{H(w,u)}{H(0,u)} − λ(w, u : D)| < ϵ.

Since the natural time-parametrization of SLE_{2} starts on the boundary of 𝕌 and goes to 0 as t → ∞, it will be useful to consider the time-reversal of the random walk Γ. Let γ = (γ_{0}, ..., γ_{l}) be the loop-erasure of this timereversal, that is γ_{0} ∈ ∂D and γ_{l} = 0. (By a simple observation, this has the same distribution of the time-reversal of the loop-erasure of Γ.) Thus, we can write Γ as γ together with a collection of loops Γ^{j}(m) := Γ(m + n_{j}), m = 0, 1, ..., n_{j−1} − n_{j}, where n_{j} := min{n ≥ 0 : Γ(n) = γ_{j}} is the hitting time of the point γ_{j} (so n_{j+1} < n_{j} for all j). If we fix j and think of D_{j} := D\ ⋃_{i=0}^{j−1}[γ_{i}, γ_{i+1}] as the new domain of the random walk, we have the following Markovian property:

Theorem 3. Conditioned on {(γ_{0}, ..., γ_{j}) = (u_{0}, ..., u_{j})}, the loops Γ^{1}, ..., Γ^{j} and the arc Γ[0, n_{j}] are independent. Furthermore, for each k, 1 ≤ k ≤ j, the conditional law of Γ^{k} is that of a simple random walk in D_{k−1} started from u_{k} and conditioned to leave D_{k−1} through the edge [u_{k}, u_{k−1}], and the conditional law of Γ[0, n_{j}] is that of a simple random walk started from 0 and conditioned to leave D_{j} at u_{j}.

One last tool we will need is the discrete Green’s function, the function G_{D}(u, v) given by the expected number of visits to v by a simple random walk started at u and stopped upon exiting D. By the above property, it follows that conditioned on the event {(γ_{0}, γ_{1}) = (u_{0}, u_{1})}, the expected number of visits to v by Γ^{1} is G(u_{1}, v)H(v, u_{1}).

We are now ready to present the core estimate of the proof of Theorem 1. Let γ_{j} and D_{j} be as above, and let ψ_{j} : D_{j} → 𝕌 be conformal maps with ψ_{j}(0) = 0 and ψ′_{j}(0) > 0. Let U_{j} = ψ_{j}(γ_{j}) be the extensions of these maps to the points γ_{j}. Note that γ is a continuously growing simple curve from ∂D to 0, and so it can be represented as a Loewner evolution. Let W : [0, ∞) → ∂𝕌 be the driving process for this evolution, so U_{j} = W (t_{j}) for some times t_{j}. (In fact, t_{j} is the continuous capacity of γ[0, j] from 0 in D.) Denote the argument of W (t)/W (0) by θ(t) (i.e., θ(t) is the unique continuous realvalued function such that θ(0) = 0 and W (t) = W (0) exp(iθ(t)) for all t.)

Let Δ_{j} = θ(t_{j}) be the argument of U_{j}/U_{0}, so U_{j} = U_{0} exp(iΔ_{j}). Then we have the following:

Theorem 4. There exists a constant C such that for all small δ > 0, there exists r_{0}(δ) such that the following holds: Let rad_{0}(D) > r_{0}. For every u_{0} ∈ V_{∂}(D) with P[γ_{0} = u_{0}] > 0, let m = min{j ≥ 1 : t_{j} ≥ δ^{2}or|Δ_{j}| ≥ δ}. Then |E[Δ_{m}]| ≤ Cδ^{3}, and |E[Δ_{m}^{2}] − 2E[t_{m}]| ≤ Cδ^{3}.

For the proof, see [1]. The idea is to consider the random quantity h_{j}^{+} :=

∑_{k=j+1}^{l} h_{k}, where each h_{k} is the number of visits to v of the loop Γ^{j}, and then

estimate both sides of the equation

E[h_{0}^{+}] = E [E[h_{0}^{+}|γ[0, m]]]

in terms of Green’s functions and hitting probabilities. These estimates then translate into analytic qualities according to Theorem 2.

The result of this estimate is that as the size of the domain goes to ∞, the numbers Δ_{m} behave ”almost” like the Brownian motion B(2t_{m}), in the sense that E[Δ_{m}] is small and E[Δ_{m}^{2}] is close to 2t_{m}. This is the first evidence for the convergence of the driving process, as well as an explanation for the constant 2 in the statement of Theorem 1.

3.3 Recognizing the driving process

In the last section, we saw that the driving function for the LERW is approximately equal to exp(iB(2t)), where B is a standard Brownian motion. In what sense this approximation holds is made more precise by the following (recall W (t) = exp(iθ(t))):

Theorem 5. For every T > 0 and ϵ > 0, there is an r_{1}(ϵ, T) such that for all D with rad_{0}(D) > r_{1}, there is a coupling of γ with Brownian motion B(t) starting at a random point in [0, 2π] such that

P [sup{|θ(t) − B(2t)| : 0 ≤ t ≤ T } > ϵ] < ϵ

The idea of the proof is to iterate the previous estimate for each step Δ_{m_{n}}, where m_{n} is defined inductively as the first j ≥ m_{n} + 1 such that |Δ_{j} − Δ_{m_{n}}| ≥ δ or t_{j} − t_{m_{n}} ≥ δ^{2}, whichever happens first. Then, let M_{n} = ∑_{j=0}^{n−1}(Δ_{m_{j+1}} − Δ_{m_{j}} − E[Δ_{m_{j+1}} − Δ_{m_{j}}|F_{j}]), where F_{n} is the σ-field generated

by γ[0, m_{n}]. Because M_{n} is a martingale with ||M_{n} − M_{n−1}||_{∞} ≤ 2δ and M_{0} = 0 a.s., the Skorohod embedding theorem then implies that we may couple (M_{0}, ..., M_{N}) with a standard Brownian motion with stopping times τ_{0} ≤ ... ≤ τ_{N} such that B_{τ_{n}} = M_{n}, and such that τ_{n+1} ≤ inf{t ≥ τ_{n} : |B_{t} − B_{τ_{n}}| ≥ 2δ}. After some more work, it follows that τ_{n} is close to 2t_{m_{n}} for all n, and so by continuity of B(t), the theorem follows.

Thus, we have a kind of convergence of LERW to SLE_{2}, but as mentioned previously, more can be said.

3.4 Convergence with respect to a stronger topology

First, we will deal with convergence of the paths of LERW and SLE_{2} in the circle.

Theorem 6. For any sequence D_{n} with rad_{0}(D_{n}) → ∞, if μ_{n} denotes the law of ψ_{D_{n}} ∘ γ^{n}, where γ^{n} is the time-reversal of LERW from 0 to ∂D_{n}, then μ_{n} converges weakly (with respect to the metric ρ defined previously) to the law of radial SLE_{2} in the unit circle.

The proof uses a tightness argument combined with the previous theorem. Basically, we define a suitable family of compact subsets of the space of simple paths, which can then be used to show that the sequence μ_{n} is tight. Therefore, a subsequence of μ_{n} converges weakly to some probability measure, which must be the law of SLE_{2}.

All that remains, then, in the proof of Theorem 1 is a translation of this convergence in the circle to convergence in D. To do this, let D_{δ} be the component of 0 in the complement of all the closed square faces in the grid δℤ^{2} intersecting ∂D. Let γ_{δ} be the time-reversal of the loop-erased random walk from 0 to ∂D_{δ}, and let β be the radial SLE_{2} path in 𝕌. Let ϕ_{δ} : 𝕌 → D_{δ} be the conformal maps with ϕ_{δ}(0) = 0 and ϕ′_{δ}(0) > 0, and let ϕ : 𝕌 → D be a similar conformal map. Then we know that with the capacity parametrization, sup{|ϕ_{δ}^{−1} ∘γ_{δ}(t)−β(t)| : t ≥ 0} → 0 in probability.

The Carathéodory kernel theorem implies that ϕ_{δ} → ϕ uniformly on compact subsets of 𝕌 as δ → 0, and so sup{γ_{δ}(t) − ϕ ∘ β(t)| : t ≥ t_{0}} → 0 in probability for all t_{0} > 0. Combined with some other estimates, this implies that for all ϵ > 0, as δ → 0,

P [sup{|γ_{δ}(t) − ϕ ∘ β(t)| : t > 0} < 3ϵrad_{0}(D)] → 1.

And since ϵ > 0 is arbitrary, this completes the proof of Theorem 1.

References

[1] Lawler, G., Schramm, O., and Werner, W. Conformal invariance of planar loop-erased random walks and uniform spanning trees Ann. Probab. 32 (2004), no.1B, 939-995;

Aubrey Clayton, UC Berkeley email: aclayton@math.berkeley.edu

4 The Loewner differential equation and slit mappings

after D. E. Marshall and S. Rohde [1] A summary written by Sunhi Choi

Abstract

We show that the Loewner equation generates slits if the driving term is Hölder continuous with exponent 1/2 and small norm, and that this is best possible.

4.1 Introduction and result

Let Ω_{t}, t_{0} ≤ t ≤ t_{1}, be a continuously increasing sequence of simply connected planar domains, and let z_{0} ∈ ∩_{t}Ω_{t}. Then there are conformal maps f_{t} : 𝔻 → Ω_{t} of the unit disc with f_{t}(0) = z_{0} and f′(0) > 0. Since f′_{t}(0) is increasing in t, we may assume that f′_{t}(0) = e^{t} for t_{0} ≤ t ≤ t_{1}. The family f_{t}, t_{0} ≤ t ≤ t_{1}, is called a Loewner chain and satisfy

∂_{t}f(z, t) = zp(z, t)∂_{z}f(z, t)  (1)

where p(z, t) is analytic in z and has positive real part. A particulary important case is the slit domains

Ω_{t} = Ω \ γ[t, t_{1}]  (2)

where Ω ⊆ ℂ is simply connected and γ is a simple arc contained in Ω except for γ(t_{1}) ∈ ∂Ω. In this case, (1) becomes

∂_{t}f = z \frac{λ(t) + z}{λ(t) − z} ∂_{z}f  (3)

where λ(t) = f_{t}^{−1}(γ(t)) ∈ 𝕋 = ∂𝔻. It is known that λ(t) is continuous, and conversely that (3) generates a sequence of increasing domains whenever λ is continuous. However, the domain Ω_{t} obtained from (3) are not necessarily slit domains if λ is only assumes to be continuous. The main result of this paper is a sharp condition on λ that guarantees slit domains.

We will first consider (2) with Ω = ℂ and t_{1} = ∞. Define a quasiarc to be the image of [0, ∞) under a quasiconformal homeomorphism of ℂ. The geometric characterization of quasiarcs is the Ahlfors three-point condition. Denote Lip(\frac{1}{2}) the space of Hölder continuous functions λ with exponent 1/2, |λ(s) − λ(t)| ≤ c|s − t|^{1/2}, and let ∥λ∥_{\frac{1}{2}} denote the smallest such c.

Theorem 1. If γ is a quasiarc, then λ ∈ Lip(\frac{1}{2}). Conversely, there is a constant C > 0 such that if λ ∈ Lip(\frac{1}{2}) with ∥λ∥_{\frac{1}{2}} < C, then ℂ \ f_{t}(𝔻) is a quasiarc for all t.

The quasiconformal constant K of ℂ\f_{t}(𝔻) depends on C only and K → 1 as C → 0. Also ∥λ∥_{\frac{1}{2}} < C can be replaced by

inf_{ϵ>0} sup_{|t−s|<ϵ} \frac{|λ(t) − λ(s)|}{\sqrt{|t − s|}} < C.

By a transformation, the Loewner equation becomes an ODE: For t ≤ s, set

ϕ_{s,t} = f_{s}^{−1} ∘ f_{t} : 𝔻 → 𝔻.

Then ϕ_{s,t}(0) = 0, ϕ′_{s,t}(0) = e^{t−s}, and (3) becomes

∂_{s}ϕ_{s,t}(z) = −ϕ_{s,t}(z)\frac{λ(s) + ϕ_{s,t}(z)}{λ(s) − ϕ_{s,t}(z)}.

Fix t and set g_{s} = ϕ_{t+s,t}, for 0 ≤ s ≤ s_{0} := t_{1} − t, and let ξ(s) = λ(t + s). Then

∂_{s}g_{s} = −g_{s} \frac{ξ(s) + g_{s}}{ξ(s) − g_{s}} for s ∈ [0, s_{0}), g_{0} = id.  (4)

Since the solutions to the initial value problem (4) are unique, it is more natural to work with (4). Theorem 1 is a simple consequence of

Theorem 2. Let ξ : [0, s_{0}] → ∂𝔻 be continuous and g_{s} the solution to (4). If g_{s_{0}}(𝔻) is a quasislit-disc, then ξ ∈ Lip(\frac{1}{2}). Conversely, if ∥ξ∥_{\frac{1}{2}} < C, then g_{s}(𝔻) is a quasislit-disc for all s.

A quasislit-disc is the image of a radially slit disc under a quasiconformal map fixing 𝔻 and 0. In section 2, we study basic properties of quasislitdisc, from which we will show that ξ ∈ Lip(\frac{1}{2}) if g_{s_{0}}(𝔻) is a quasislit-disc. To prove the converse, we will approximate ξ ∈ Lip(\frac{1}{2}) by functions ξ^{(ϵ)} ∈ Lip(\frac{1}{2}) for which the solutions g_{s}^{(ϵ)} generates slits. If ∥ξ^{(ϵ)}∥_{\frac{1}{2}} is small, then the conformal welding in these slits can be shown to be quasisymmetric, and which implies that the slits are quasiconformal slits. By compactness of the space of quasislit domains, the limits are also quasislit-discs.

4.2 Slit discs

Let Γ ⊂ 𝔻̄ \ {0} be a Jordan arc with endpoints a ∈ 𝕋 and b ∈ 𝔻 \ {0}. We call 𝔻 \ Γ, a slit-disc with initial point a and tip b. Define the conformal welding on a slit disc D as follows. Let g : 𝔻 → D be conformal with g(0) = 0 and g(1) = b. Then there are two preimages α^{+}, α^{−} ∈ 𝕋 of the initial point a. Let I be the arc < α^{−}, α^{+} > on 𝕋, which contains 1. Then the welding homeomorphism ϕ : I → I is the unique continuous function with ϕ(α^{+}) = α^{−} and

g(x) = g(ϕ(x)) for all x ∈ I.

In Lemma 3 we show that conformal welding in slits is quasisymmetric if and only if the slits are quasiconformal slits.

Lemma 3. 𝔻 \ Γ is a quasislit-disc if and only if there is a constant 1 ≤ M < ∞ such that

M^{−1} ≤ \frac{|ϕ(x) − 1|}{|x − 1|} ≤ M

for all x ∈< 1, α^{+} > and

M^{−1} ≤ \frac{|ϕ(x) − ϕ(y)|}{|ϕ(y) − ϕ(z)|} ≤ M

whenever x, y, z ∈< 1, α^{+} > with |x − y| = |y − z|. The quasiconformal constant K of 𝔻 \ Γ depends on M only, with K → 1 if and only if M → 1.

The slit of a quasislit disc is a quasiarc, whereas the converse is not true. In Lemma 4 we give a geometric characterization which guarantees a quasiarc to be the slit of a quasislit-disc.

Lemma 4. The slit disc 𝔻\Γ is a quasislit-disc if and only if Γ is a quasiarc that approaches 𝕋 nontangentially.

The proof of the Hölder continuity of the Loewner parameter for quasislitdiscs follows from Lemma 5 and Lemma 6.

Lemma 5. Let D = 𝔻 \ Γ be a K-quasislit-disc with initial point a and tip b. Let c and d be points on Γ such that the arc Γ_{c} of Γ from c to a contains Γ_{d}. Let ϕ_{c}, ϕ_{d} be conformal maps of 𝔻 onto 𝔻 \ Γ_{c}, 𝔻 \ Γ_{d} fixing 0. Then ϕ_{d}^{−1}(ϕ_{c}(𝔻)) is a K-quasislit-disc, too.

Lemma 6. Let D = 𝔻\Γ be a K-quasislit-disc with initial point a and tip b. Let f : 𝔻 → D be the conformal map fixing 0 and assume r := f′(0) > 1/2. Then

|f^{−1}(b) − a| ≤ C\sqrt{1 − r}

where C depends on K and C(1) = 0.

To prove the converse, we need to construct quasislit-discs with prescribed a, f^{−1}(b) and f′(0).

Lemma 7. Given a, a′ ∈ 𝕋 and 0 < r < 1, there is a slit disc D = D_{a,a′,r} with initial point a and tip b such that the conformal mal f : 𝔻 → D with f(0) = 0 satisfies

f′(0) = r and f(a′) = b.

Moreover, for r ≥ 1/2, D_{a,a′,r} can be chosen as a K-quasislit disc with

K ≤ C(\frac{|f^{−1}(b) − a|}{\sqrt{1 − r}})

where C : [0, ∞) → [1, ∞) is continuous with C(0) = 1.

Finally, we will prove a partial converse of Lemma 5. Let D_{i}, 1 ≤ i ≤ n be slit-discs with initial points a_{i} and tips b_{i}. Define the conformal concatination

D_{1} ∗ D_{2} ∗ ... ∗ D_{n} = g_{n} ∘ ... ∘ g_{2} ∘ g_{1}(𝔻),

where g_{i} : 𝔻 → D_{i} are conformal maps such that g_{i}(a_{i−1}) = b_{i} and g_{i}(0) = 0. the concatination of quasislit discs is a quasislit disc and the constant of the concatination can be bounded independently of n, if |g′_{i}(0)| is bounded away from 1.

Lemma 8. If D_{i} are K-quasislit discs and |g′_{i}(0)| ≤ 1 − ϵ for all 1 ≤ i ≤ n, then D_{1} ∗ ... ∗ D_{n} is a K′-quasislit disc with K′ depending only on ϵ and K and K′ → 1 as K → 1 with ϵ fixed.

4.3 The Loewner equation and conformal welding

Let ξ : [0, s_{0}] → 𝕋 be continuous and let g_{s} be the solution to (4). For z ∈ 𝕋 denote

T (z) = sup{s > 0 : g_{t}(z) ≠ ξ(t) for all t ∈ [0, s]}.

If T (z) < ∞ we have g_{T (z)}(z) = ξ(T (z)), and for the conformal welding ϕ : I → I,

T (z) = T (z′) if and only if ϕ(z) = z′ or z = z′.

In the special case ξ(t) = ξ_{0}(t) ≡ 1, calculation shows

T_{0}(e^{iφ}) = − log \frac{1 + cos φ}{2}.

In Lemma 9, we show that pairs of points z, z′ hit the singularity in finite time if ∥ξ∥_{\frac{1}{2}} is small

Lemma 9. There is a constant C_{0} such that if ∥ξ∥_{\frac{1}{2}} < C_{0}, then

C^{−1} ≤ \frac{T (z)}{T_{0}(z)} ≤ C

for all z ∈ 𝕋 with ℜz > 0, where C = C(∥ξ∥_{\frac{1}{2}} ) → 1 as ∥ξ∥_{\frac{1}{2}} → 0.

Lemma 10. There is a constant C_{0} such that if ∥ξ∥_{\frac{1}{2}} < C_{0}, z_{1}, z_{2}, z_{3} ∈< 1, α^{+} > ∩{ℜz > 0} and

|z_{1} − z_{2}| = |z_{2} − z_{3}|,

then

C^{−1} ≤ \frac{|ϕ(z_{1}) − ϕ(z_{2})|}{|ϕ(z_{2}) − ϕ(z_{3})|} ≤ C

where C → 1 as ∥ξ∥_{\frac{1}{2}} → 0.

4.4 Proof of the Theorem

We decompose the Loewner equation into infinitesimal conformal factors. If g_{s} solves (4), 0 ≤ s ≤ s_{0}, then

f_{t} = g_{s_{0}} ∘ g_{s_{0}+t}^{−1} , − s_{0} ≤ t ≤ 0

is a Loewner chain satisfying (3). Note that f_{−s_{0}} = g_{s_{0}}, f_{0} = id, and λ(t) = ξ(s_{0} + t). Set

ϕ_{s,t} = f_{s}^{−1} ∘ f_{t}.

For −s_{0} = t_{0} < t_{1} < ... < t_{n} = 0 we have

g_{s_{0}} = f_{−s_{0}} = ϕ^{(n)} ∘ ϕ^{(n−1)} ∘ ... ∘ ϕ^{(1)} , ϕ^{(j)} = ϕ_{t_{j},t_{j−1}}.

The ϕ^{(j)} are the solutions to

∂_{s}ϕ_{s}^{(j)}(z) = −ϕ_{s}^{(j)}(z)\frac{λ_{j}(s) + ϕ_{s}^{(j)}(z)}{λ_{j}(s) − ϕ_{s}^{(j)}(z)}, λ_{j}(s) = λ_{j}(t_{j−1} + s), ϕ_{0}^{(j)} = id. (5)

with 0 ≤ s ≤ t_{j} − t_{j−1}. Conversely, if solutions ϕ^{(j)} to (5) are given with λ_{j−1}(t_{j} − t_{j−1}) = λ_{j}(0), then the solutions f_{−s_{0}} resp. g_{s_{0}} to (3) resp. (4) are ϕ^{(n)} ∘ ϕ^{(n−1)} ∘ ... ∘ ϕ^{(1)}. In particular, if all ϕ^{(j)} map to slit discs, then f_{t} and g_{s} map to slit discs.

Proof of Theorem 2: First assume that g_{s_{0}}(𝔻) is a K-quasislit disc. Applying Lemma 5 and Lemma 6 for f = f_{s}^{−1} ∘ f_{t} = g_{s+s_{0}} ∘ g_{t+s_{0}}^{−1}, we can show that ξ ∈ Lip(\frac{1}{2}).

Now let ξ : [0, s_{0}) → 𝕋 with small ∥ξ∥_{\frac{1}{2}} be given. In order to approximate ξ by functions ξ_{n} ∈ Lip(\frac{1}{2}), set

t_{j} = \frac{j}{n}s_{0} , ξ_{j} = ξ(t_{j}) , r = e^{− \frac{s_{0}}{n}}

for 0 ≤ j ≤ n, and let

D_{j} = D_{ξ_{j+1},ξ_{j},r}

be the slit disc provided by Lemma 7. Then the normalized conformal map ϕ^{(j)} : 𝔻 → D_{j} maps ξ_{j} to the tip of D_{j}, and ξ_{j+1} is the base. Let λ_{j} be the Loewner parameter of D_{j}, then λ_{j}(0) = ξ_{j}, λ_{j}(\frac{s_{0}}{n}) = ξ_{j+1}, and the composition

g_{s_{0}}^{(n)} = ϕ^{(n−1)} ∘ ... ∘ ϕ^{(0)}

is the solution to (3) with parameter ξ_{n} given by

ξ_{n}|_{[t_{j},t_{j+1}]}(t) = λ_{j}(t − t_{j}).

From Lemmas 3, 9, and 10 it follows that D(n) = g_{s_{0}}^{(n)}(𝔻) is a quasislit disc. g_{s}^{(n)} converges to a conformal map g_{s} onto a quasislit disc and since ξ_{n} converges to ξ, we conclude that g_{s} also satisfy (4) and Theorem 2 follows. Theorem 1 can be derived from Theorem 2.

References

[1] Marshall, D.E. and Rohde, S. The Loewner differential equation and slit mappings. preprint.

Sunhi Choi, MIT email: schoi@math.mit.edu

5 Aggregation in the Plane and Loewner’s Equation

after L. Carleson and N. Makarov [1] A summary written by Nick Crawford

Abstract

The goal of this summary is twofold. First to discuss the general Loenwer equation. we shall see that SLE but one of many types of growth processes that can be generated. In this respect, we will attempt to give a more filled out version of the introduction in [1]. The second goal is to describe a general method appearing in section 3 of

[1] which lays out a general method for obtaining growth estimates on certain types of evolving processes. The examples of Loewner Chains given in the second section of the present summary provide the motivating example.

5.1 An Introduction to Loewner’s Equation

Following [1], assume given a chain K_{t} of compact subsets of the plane satisfying K_{s} ⊂ K_{t}, s < t. Without loss of generality, and for simplicity, we assume K_{0} = 𝔻̄. Let Δ = ℂ̂\𝔻̄. By the Riemann Mapping theorem, for each t there exists a (unique) mapping ϕ_{t} : Δ → ℂ̂\K_{t} satisfying ∞ ↦ ∞ and ϕ′_{t}(∞) > 0. We would like to be able to analyze the trajectories t ↦ ϕ_{t}(z). We can differentiate with respect to t any trajectory that is absolutely continuous. In fact, t ↦ ϕ′_{t}(∞) is absolutely continuous function of t is enough to imply that all trajectories are absolutely continuous. Once we have differentiability, another consequence of K_{s} ⊂ K_{t} is that

ℜ[\frac{∂_{t}ϕ(t, z)}{zϕ′(t, z)}] > 0.  (1)

To explain the first statement, let us consider ψ_{t}(w) = ϕ_{t}(1/w). The inclusions K_{s} ⊂ K_{t} imply that there is a mapping ψ(z, s, t) : 𝔻 ↪ 𝔻, normalized so that ψ(0, s, t) = 0 and ψ′(0, s, t) > 0 and ψ_{t} = ψ_{s}(ψ(z, s, t).

In this case,

|ψ_{s}(z) − ψ_{t}(z)| = | ∫_{ψ(z,s,t)}^{z} ψ′_{s}(ζ)dζ|.  (2)

By distortion, |ψ_{s}(z)−ψ_{t}(z)| ≤ |1−\frac{ψ(z,s,t)}{z}|\frac{2}{(1−r)^{3}} ϕ′_{s}(∞). Now ℜ[1−\frac{ψ(z,s,t)}{z}] > 0 by the Schwarz lemma, so the Herglotz representation (mentioned below) implies |1 − \frac{ψ(z,s,t)}{z}| ≤ \frac{1+r}{1−r}(1 − \frac{ϕ′_{t}(∞)}{ϕ′_{s}(∞)}).

Loewner’s Theory implies that one may go in the reverse direction. In particular, suppose A(t, z) are a one parameter family of analytic functions on Δ satisfying ℜ(\frac{A(t,z)}{z}) > 0, A′(t, ∞) > 0. First, one may use the Herglotz representation theorem to write

\frac{A(t, z)}{z} = ∮ \frac{z + ξ}{z − ξ} dμ_{t}(ξ)  (3)

for some positive measure μ_{t} on 𝕋.

To find such a μ_{t} is not difficult: define

f_{ϵ}(E, z) = ∮ ℜ(\frac{z + ξ}{z − ξ})ℜ(\frac{A(t, (1 − ϵ)ξ)}{(1 − ϵ)ξ})χ_{E}(ξ)dξ.  (4)

Harnak’s inequality implies the totality of this collection of positive harmonic functions is a normal family. By diagonalization, we can take limits for the collection of E = [r_{1}, r_{2}] with r_{i} ∈ ℚ/ℤ. Evaluating the limit collection at ∞ then defines the measure μ_{t}. Another way of saying this is that we consider a weak limit point among the measures on the unit circle

{f_{ϵ}(ξ)dξ}.

Forming the differential equation

∂_{t}ϕ_{t}(z) = ϕ′(t, z)A(t, z),  (5)

the Herglotz representation implies that if ∥μ_{t}∥ ∈ L_{loc}^{1}(𝕋), then so is A(t, z), so an analytic solution to (5) exists for any (analytic) initial condition and will be absolutely continuous in t. Moreover all conditions stated in the first paragraphs of this section will be satisfied for this chain of maps.

Notice the following relation, which will be relevant in the next sections. If

c(t) := ϕ′_{t}(∞),  (6)

then c(t) satisfies

∂_{t}c(t) = ∥μ_{t}∥c(t).  (7)

As ϕ′(∞) is closely related to the diameter of K_{t} we make the following definition alluded to in the abstract (see [2]).

Definition 1. T, the doubling time of growth process associated to μ_{t} is defined by ∫_{0}^{T} ∥mu_{t}∥dt = 1.

One last important point: Loewner’s Equation, a PDE in t, z can be viewed as an ODE if one considers the inverse evolution. In particular, let

w(s, t, z) = ϕ_{s}^{−1}(ϕ_{t}(z)).  (8)

Then a calculation using (5) shows that w satisfies

∂_{s}w = −A(s, w),  (9)

with initial condition w(t, t, z) = z.

5.2 A Continuous time/space analog of DLA and a Deterministic Counterpart

The authors were motivated by Kesten’s result on the doubling time of DLA to study an analog in terms of Loewner’s Equation. As mentioned above, the doubling time for the capacity is roughly the same as the doubling time for the diameter, making capacity an appropriate replacement for size in a continuous analog of Kesten’s Theorem.

Consider the following probabilistic example of Loewner’s equation. Let τ_{j} be the sequence arrival times of a Poisson process of rate δ. For each τ_{j} choose ξ(τ_{j}) uniformly at random from ∂Δ. Define the (random) growth measures μ_{t} by

∫_{0}^{t} μ_{s}ds = ∑_{τ_{j}≤t} δ_{ξ(τ_{j})},  (10)

where δ_{ξ} is a Dirac measure. This growth process can be seen as a continuous time/space analog to DLA: Basically, the growing set these measures create is a sequence of geodesic ’hairs’ of length approximately δ growing out of the disc. At each arrival of the Poisson process, we are asked to choose a new point on the outer boundary of K_{t} at random according it’s harmonic measure and grow for a length of time which increases the capacity of the shape by δ, give or take.

We now give an example of deterministic DLA. It is related to the previous example (see [1] for an explanation of how). Suppose {ϕ_{t}} is a general chain of conformal mappings. Fix δ > 0. For each ξ ∈ ∂Δ define

ϵ_{t} = ϵ_{t}(ξ, δ) = inf{ϵ > 0 : d(ϕ_{t}((1 + ϵ)ξ), Ω_{t}) = δ}.  (11)

Definition 2. A Hele-Shaw δ process {ϕ_{t}} is a Loewner chain such that μ_{t} satisfies dμ_{t}(ξ) = ϵ_{t}^{2}(ξ, δ)d(ξ).

5.3 A Method for Estimating the Growth Time

Let us consider a Loewner process such that the analytic maps are defined on the upper-half plane (with the same normalizations as before, i.e. ϕ(∞) = ∞, ϕ′(∞) > 0)and assume ϕ_{0}(z) = z. Let μ_{t} be a locally integrable chain of finite measures on ℝ. The Loewner differential equation then takes the same form as above if we change the potential:

A(t, z) = ∫ \frac{dμ_{t}ξ}{ξ − z} .  (12)

We shall, from now on, work with the inverse Loewner equation. Differentiating (9), we have upon integrating on [0, t],

log |ϕ′_{t}(z)| = ∫_{0}^{t} ℜ(A′(s, w(s, t, z))ds.  (13)

Similarly,

log \frac{ℑ[ϕ_{t}(z)]}{ℑz} = ∫_{0}^{t} ∫_{ℝ} \frac{dμ_{s}(ξ)}{(x − ξ)^{2} + y^{2}} ds.  (14)

Fix δ > 0. Consider ρ(t, x) such that d(ϕ_{t}(x+iρ), ∂Ω_{t}) ≍ δ. An application of the distortion theorem shows this to be equivalent to ρ|ϕ′_{t}(x + iρ)| ≍ δ. In this case, combining the previous two equations, we have

log \frac{δℑ[ϕ_{t}(z)]}{ρℑz} = ∫_{0}^{t} ∫_{ℝ} \frac{(x − ξ)^{2}dμ_{s}(ξ)}{[(x − ξ)^{2} + y^{2}]^{2}} ds.  (15)

Note that this occurs for an HS(δ) process with ρ = ϵ(t, x). This motivates the general theorem we now present.

Let {μ_{t}} be a family of finite positive measures defined on 𝕋, extended periodically to ℝ. For the time being we do not assume the μ_{t} come from Loewner’s equation. This collection induces a measure, denoted by μ, on

Ω := [0, T] × 𝕋.  (16)

Another natural measure associated to this space is the Lebesgue product measure, λ.

For each x ∈ 𝕋, define

M_{x}(s, y) := ∫_{ℝ} \frac{dμ_{s}(ξ)}{(x − ξ)^{2} + y^{2}} ,  (17)

K_{x}(s, y) := ∫_{ℝ} \frac{(x − ξ)^{2}dμ_{s}(ξ)}{[(x − ξ)^{2} + y^{2}]^{2}} ,  (18)

with 0 ≤ s ≤ T and y > 0, and consider the family of solutions to the differential equation

\frac{∂_{s}y}{y} = −M_{x}(s, y).  (19)

Every positive function ρ(ω) on Ω defines an initial condition for each t, x ∈ Ω by

ρ(ω) = y_{ω}(t).  (20)

We note that the solution y_{ω}(·) exists on 0 ≤ s ≤ t, and denote Y (ω) = y_{ω}(0). In other words, ρ can be thought of as a random initial height and Y is the resulting final height of the process.

Theorem 3. Let μ be a probability measure and ρ be a positive function on Ω. Suppose there exists δ ∈ (0, 1) such that

log \frac{δY (ω)}{ρ^{2}(ω)} ≥ ∫_{0}^{t} K_{x}(s, y_{ω}(s))ds.  (21)

Then there exists c > 0 such that either:

μ\{ρ ≤ δ^{\frac{1+c}{2}} \} > \frac{1}{2}  (22)

or we can find Ω′ ⊂ Ω such that

μ(Ω′) > c, λ(Ω′) < δ^{c}λ(Ω).  (23)

The authors give another result which provides a much sharper bound for the HS(δ) process. However, the theorem stated here, which will be the focus of the talks in Catalina, can be applied to many more types of evolution processes.

References

[1] Carleson, L. and Makarov, N., Aggregation in the Plane and Loewner’s Equation. Commun. in Math. Phys., 216 (2001), pp. 583–607;

[2] Pomerenke, Ch. Univalent Functions Vandenhoeck and Ruprecht (1975).

Nick Crawford, UCLA email: crawford@math.ucla.edu

6 Conformal restriction: the chordal case

after G. Lawler, O. Schramm, and W. Werner [1] A summary written by Jesse Goodman

Abstract

The paper characterizes the law of chordal random subsets K of ℍH (connecting 0 and ∞) that satisfy the conformal restriction property: conditioned on avoiding certain subsets A ⊂ ℍ, the law of K is the same as the law of Φ^{−1}(K) for a conformal map Φ : ℍ\A → ℍ fixing 0 and ∞. The law of such a K comes from a one-parameter family closely linked to the stochastic Loewner evolution with κ ≤ 8/3. A related property, one-sided restriction, is developed and is shown to be related to the processes SLE(κ, ρ) and to the right boundaries of Brownian excursions and reflected Brownian excursions.

6.1 Preliminaries

The conformal restriction property involves the law of K conditioned on avoiding certain sets. We will begin with some definitions about what sort of sets K might be expected to avoid.

A bounded hull is a compact subset A ⊂ \overline{ℍ} such that A = \overline{A ∩ ℍ} and such that ℍ\A is simply connected (and connected). The set of bounded hulls is written Q. Bounded hulls not containing 0 are called ∗-hulls and the set of them is Q^{∗}. (It is ∗-hulls that will be the sets K should avoid.) Q_{+} is the set of hulls for which A ∩ ℝ ⊂ (0, ∞).

For A ∈ Q, there exists a unique conformal map g_{A} : ℍ\A → ℍ with the normalization g_{A}(z) = z + O(1/z) as z → ∞. For A ∈ Q^{∗}, again define Φ_{A} to be a conformal map Φ_{A} : ℍ\A → ℍ but this time require that Φ(0) = 0, Φ(∞) = ∞, and Φ(z) = z + O(1) as z → ∞. The g_{A} will be relevant to Loewner’s differential equation, while the Φ_{A} are the appropriate maps when changing domains since they fix the distinguished points 0 and ∞.

6.2 The two-sided restriction measures

The (two-sided) restriction measures P_{α} will be the law of a subset K ⊂ ℍ with the properties:

1. \overline{K} = K ∪ \{0\}.

2. K is connected and ℂ\\overline{K} is connected.

Accordingly, define Ω to be the set of all such subsets K. The Hausdorff metric on closed subsets of \overline{ℍ} ∪ \{∞\} induces the Borel σ-field on Ω. It is noted (without proof) that this σ-field is in fact generated by events of the form {K ∈ Ω : K ∩A = ∅} for A ∈ Q^{∗}. It follows that a probability measure P on Ω will be uniquely determined by the probabilities P (K ∩ A = ∅) as A ranges over Q^{∗} (Lemma 3.2 of [1]).

Definition 1. A measure P on Ω is a (two-sided) restriction measure if it is the law of a random subset K such that:

1. The law of K is scale-invariant; and

2. Conditional on not intersecting A ∈ Q^{∗}, the random subset K has the same distribution as Φ_{A}^{−1}(K′), where K′ is a copy of K without any conditioning.

Proposition 2. (Proposition 3.3 of [1]) These properties uniquely determine P = P_{α} up to a single parameter α ≥ 0, and for A ∈ Q^{∗},

P_{α}(K ∩ A = ∅) = Φ′_{A}(0)^{α}

The proof relies on the fact that {Φ_{A} : A ∈ Q^{∗}} forms a semigroup under composition; this induces a semigroup structure on Q^{∗} as well, defined by Φ_{A·A′} =^{def} Φ_{A} ∘ Φ_{A′}. Reformulated with this terminology, condition 2 of definition 1 states that

P (K ∩ (A · A′) = ∅) = P (K ∩ A = ∅)P (K ∩ A′ = ∅)

i.e., F (A) =^{def} P (K ∩ A = ∅) is a semigroup homomorphism. The idea of the proof is to precompose F with a semigroup homomorphism G defined on [0, ∞), giving a semigroup homomorphism F ∘ G : [0, ∞) → [0, 1]. Such homomorphisms can be characterized, giving the parameter α.

Note that this proposition does not prove the existence of P_{α}, and indeed the P_{α} only exist for α ≥ 5/8 (Corollary 8.6 of [1]). For these values of α, the P_{α} can be directly constructed in terms of SLE and Brownian motion.

Theorem 3. (Theorem 6.1 of [1]) The law of the SLE_{8/3} path γ starting at 0 is P_{5/8} and

P (γ[0, ∞) ∩ A = ∅) = Φ′_{A}(0)^{5/8}

for all A ∈ Q^{∗}.

Theorem 4. (Proposition 4.1 of [1], following Bálint Virág [2]) The law of filling of a Brownian excursion B in ℍ (i.e., a Brownian excursion with all the “holes” filled in) is P_{1} and equivalently

P (B[0, ∞) ∩ A = ∅) = Φ′_{A}(0)

for every A ∈ Q^{∗}.

Theorem 5. (see Theorem 7.3 of [1]) For any α ≥ 5/8, let

κ = \frac{6}{2α + 1}, λ = \frac{(8 − 3κ)(6 − κ)}{2κ} = 2α(8α − 5),

and let Ξ be the filling of an SLE_{κ} curve together with a cloud of Brownian bubbles with intensity λ. (See [1], p.29 for the specifics of how this is done.) Then the law of Ξ is P_{α}.

Observe also that if K_{i} has law P_{α_{i}} for i = 1, …, n, then the law of the filling of K_{1} ∪ ⋯ ∪ K_{n} is P_{α} where α = α_{1} + ⋯ + α_{n}. As a consequence, the filling of 5 Brownian excursions is equal in law to the filling of 8 SLE_{8/3}, both having law P_{5}.

Theorem 5 gives a second interpretation of P_{1}. α = 1 gives κ = 2, so a Brownian excursion is equivalent to an SLE_{2} path with Brownian bubbles added (with some intensity λ > 0). This is quite reasonable, since Brownian motion is the scaling limit of random walk while SLE_{2} is the scaling limit of loop-erased random walk. So, roughly speaking, a Brownian excursion is loop-erased random walk with the loops “put back in”.

Theorem 3 is proved by considering a certain local martingale, whose initial value is Φ′_{A}(0)^{5/8} and which converges to 1_{K∩A=∅} as t increases to T = inf{t : K_{t} ∩ A ≠ ∅}. Similarly, Theorem 5 is proved by considering a more general martingale, this time starting from Φ′_{A}(0)^{α} for α ≥ 5/8, which involves a factor coming from an SLE_{κ} computation and another factor in the form of an integral. It is this factor that corresponds to the Brownian bubbles added in to produce Ξ and P_{α}.

6.3 One-sided restriction

The two-sided restriction measures do not exist for α < 5/8. Intuitively speaking, lower values of α mean that P_{α} increasingly pushes K away from ℝ\{0} (since K is less likely to intersect hulls starting from ℝ\{0} for smaller α). By α = 5/8, this two-sided pressure has already forced K to be (the trace of) a simple curve, and it cannot shrink any further. The problem is resolved by considering fewer hulls and ignoring the “left hand side” of K.

Definition 6. Q_{+} is the set of hulls A with A ∩ ℝ ⊂ (0, ∞). Ω_{+} is the set of closed connected K ⊂ ℍ such that K ∩ ℝ = (−∞, 0] and ℍ\K is connected. The natural σ-field on Ω_{+} is generated by events {K ∩ A = ∅} for A ∈ Q_{+}.

Definition 7. A measure P on Ω_{+} is a right-sided restriction measure if it is the law of a subset K such that

1. The law of K is scale invariant; and

2. Conditional on not intersecting A ∈ Q^{∗}, the random subset K has the same distribution as Φ_{A}^{−1}(K′), where K′ is a copy of K without any conditioning.

As with the two-sided restriction measures, one-sided restriction measures are uniquely characterized as P = P_{α}^{+} by a parameter α ≥ 0, with

P_{α}^{+}(K ∩ A = ∅) = Φ′_{A}(0)^{α}

for all A ∈ Q_{+}. However, one-sided restriction measures exist for all α ≥ 0. (The case α = 0 is trivial.)

Note that if K_{2} ∈ Ω, then ℍ\K has exactly two components, the left part and the right part. If the law of K_{2} is P_{α}, then the law of K_{1} is P_{α}^{+}, where K_{1} is obtained by filling in to the left of K_{2} (i.e., taking the union of K_{2} with the component of \overline{ℍ}\\overline{K_{2}} that contains (−∞, 0); this operation is denoted F_{ℍ}^{ℝ_{+}}).

Two explicit constructions of the right-sided restriction measures are given by

Theorem 8. If K_{t} is the evolving hull of the SLE(8/3, ρ) process, K_{∞} =^{def} ⋃_{t≥0} K_{t}, and K = F_{ℍ}^{ℝ_{+}}(\overline{K_{∞}}), then the law of K is P_{α}^{+}, where α and ρ are related by

α = \frac{(3ρ + 10)(2 + ρ)}{32}, ρ = \frac{−8 + \sqrt{24α + 1}}{3}, α > 5/8, ρ > −2

Theorem 9. F_{ℍ}^{ℝ_{+}} applied to a Brownian excursion that is conditioned to avoid (0, ∞) and reflected off (−∞, 0] with reflection angle θ_{1} = π(1 − α_{1}) has law P_{α_{1}}^{+}. F_{ℍ}^{ℝ_{+}} applied to the union of independent reflected Brownian excursions with θ_{i} = π(1 − α_{i}), where α_{i} ∈ (0, 1] for i = 1, …, k, has law P_{α}^{+} with α = α_{1} + ⋯ + α_{k}.

The construction of a reflected Brownian excursion (RBE) is given in Section 8.2 of [1].

SLE(κ, ρ) is a generalization of SLE_{κ}; it reduces to ordinary SLE in the case ρ = 0. As for SLE, it is given in terms of Loewner’s differential equation, driven by

W_{t} = \sqrt{κ}Y_{t} − 2 ∫_{0}^{t} \frac{du}{\sqrt{κ}Y_{u}}

where Y is a 1 + \frac{2(ρ + 2)}{κ}-dimensional Bessel process.

SLE(κ, ρ) has many of the same properties as SLE(κ), as given by Lemma 8.3 of [1]:

1. (K_{λt})_{t≥0} has the same distribution as (\sqrt{λ}K_{t})_{t≥0} for λ > 0. In particular, K_{∞} =^{def} ⋃_{t≥0} K_{t} is scale-invariant.

2. If k ≤ 4 and ρ ≥ −2 + κ/2 then K_{∞} ∩ ℝ = {0} a.s.

3. If k ≤ 4 and ρ < −2 + κ/2 then K_{∞} ∩ ℝ = (−∞, 0] a.s.

4. K_{∞} is unbounded a.s.

The two methods of constructing P_{α}^{+} lead to several equalities in law (see Corollaries 8.5 and 8.2 of [1]) between the following:

1. The right boundary of a Brownian excursion, and SLE(8/3, 2/3) (i.e., its path);

2. The right boundary of the union of two Brownian excursions, and SLE(8/3, 2);

3. The right boundary of the union of independent RBEs with angles π−θ_{i}, and the right boundary of an RBE with angle π−∑ θ_{i}, provided ∑ θ_{i} < π; 4. The right boundary of the union of 2 independent reflected Brownian

excursions with angle π/2 (i.e., orthogonally reflected off (−∞, 0]), and a Brownian excursion

5. The right boundary of an RBE with angle 3π/8, and (the path of) SLE_{8/3}

The proof of Theorem 8, like that of Theorem 5, proceeds by tracking the appropriate local martingale.

6.4 SLE_{6} and Brownian motion

The paper discusses SLE_{6} and gives a short proof of the locality property (Proposition 5.1 of [1]) in the course of proving Theorem 3. It also presents some previous results about SLE_{6} and hulls of planar Brownian motion. Let K_{t}^{fp} denote the evolving hull of full-plane SLE_{6} and X̂_{t} the hull of planar Brownian motion (i.e., the trace up to time t with the holes filled in).

Theorem 10. For every simply connected proper sub-domain D ⊊ ℂ containing the origin, let T_{D} and τ_{D} be the hitting times of ∂D by K_{t} and X̂_{t} respectively. Then K_{T_{D}} and X̂_{τ_{D}} have the same law.

Important to the proof is the fact that K_{T_{D′}} ∩ ∂D′ has the same distribution as X̂_{τ_{D′}} ∩ ∂D′ for every D′ satisfying the same conditions as D. Another version of the theorem relates chordal SLE_{6} to the hull B̂_{t} of a Brownian motion in ℍ which is started from 0 and reflected off of ∂ℍ at an angle of 2π/3 pointing away from the origin:

Theorem 11. Let A ⊂ \overline{ℍ} be a closed subset not containing 0 such that the component of \overline{ℍ}\A that contains 0 is bounded (i.e., A disconnects 0 from ∞). Then K_{T} and B̂_{τ} have the same law, where T and τ are the respective hitting times of A by K_{t} and B̂_{t}.

Theorems 10 and 11 are generalized in Theorem 9.4 of [1]. Repeated application of Theorem 11 and conformal invariance shows that the SLE_{6} path can be approximated uniformly by a Brownian motion reflected off “snapshots” of its past hull with angle 2π/3 (Corollary 9.5 of [1]). So, roughly, SLE_{6} can be interpreted as Brownian motion reflected off its past hull with angle 2π/3.

References

[1] Lawler, G., Schramm, O. and Werner, W., Conformal restriction: the chordal case, J. Amer. Math. Soc. 16 (2003), no. 4, 917–955

[2] Virág, B., Brownian beads, Probab. Theory Related Fields 127 (2003), no. 3, 367–387

Jesse Goodman, University of British Columbia email: jgoodman@math.ubc.ca

7 Basic Properties of SLE

after S. Rohde and O. Scramm [3] A summary written by Jon Handy

Abstract

We summarize the proof that the SLE trace path is a simple curve for κ in [0,4], a self-intersecting curve for κ in (4,8), and a space filling curve when κ > 8. It is moreover indicated how the SLE trace is transient.

This is a continuation of the previous section, and we make use of the notation and concepts found therein.

Let us first remark briefly on the following theorem.

Theorem 1. If κ ≠ 4, there is h(κ) > 0 such that for every bounded set A ⊂ ℍ and every t > 0, a.s. f̂_{t} is Hölder continuous with exponent h(κ) on A,

|f̂_{t}(z) − f̂_{t}(z′)| ≤ C|z − z′|^{h(κ)} ∀z, z′ ∈ A,

where C = C(ω, t, A) is random and may depend on t and A.

This is essentially proven by the method of Theorem 4 of the previous section with clever choice of the exponent b from Corollary 3 of the previous section. Our main reason for mentioning this is the following consequence, which will be needed later.

Corollary 2. For κ ≠ 4 and every t, dim_{H} ∂K_{t} < 2 a.s.

The corollary follows from the fact, proven in [1] and [2], that the boundary of the image of a disc under a Hölder continuous conformal map has Huasdorff dimension bounded away from 2.

We say z ∈ ℍ is swallowed by the SLE if z ∉ γ[0, ∞) but z ∈ K_{t} for some t ∈ (0, ∞). The smallest such t will be called the swallowing time of z.

Let γ be the SLE_{κ} trace (which we know to be a continuous path). Our primary goal is to consider the proofs of the following facts:

i If κ ∈ [0, 4] then γ is a.s. a simple path and γ[0, ∞) ⊂ ℍ ∪ {0}. ii If κ ∈ (4, 8) and z ∈ \overline{ℍ} \ \{0\} then a.s. z is swallowed.

iii If κ > 8 then a.s. γ[0, ∞) = \overline{ℍ}, i.e. γ is a space filling curve.

Geometrically, (ii) should be understood to mean that any fixed z ∈ \overline{ℍ} \ \{0\} is almost surely not on the SLE trace, but that the trace intersects itself to form loops which almost surely cover the entire upper half-plane.

To establish (i), it suffices to prove:

Lemma 3. If κ ∈ [0, 4] and γ is the SLE_{κ} trace then a.s.

γ[0, ∞) ⊂ ℍ ∪ {0}.

To understand why this suffices, we recall that if we define

ĝ_{t}(z) := g_{t+s} ∘ g_{s}^{−1}(z + ξ(s)) − ξ(s),

then {ĝ_{t}(z)}_{t≥0} =^{d} {g_{t}(z)}, so if γ̂ is the trace for {ĝ_{t} then γ̂[0, ∞) ⊂ ℍ ∪ {0}, whence

γ[s, ∞) ∩ (ℝ ∪ K_{s}) = {γ(s)} a.s.

by applying g_{s}^{−1}.

The lemma itself is dealt with by noting that Y_{x}(t) = κ^{− \frac{1}{2}} (g_{t}(z) − ξ(t)) is a Bessel process of dimension α = 1 + \frac{4}{κ}, i.e.

dY_{x}(t) = \frac{\frac{1}{2}(α − 1)}{Y_{x}(t)} dt + dB_{t},

where B_{t} is a standard Brownian motion. The theory of Bessel processes provides that a.s. Y_{x}(t) ≠ 0 for t ≥ 0, x ≠ 0, and κ ∈ [0, 4]; i.e. x ∉ K_{t}, providing the lemma.

Let us recall the definition of the hypergeometric function in 𝔻,

2F_{1}(η_{0}, η_{1}, η_{2}, z) = ∑_{n=0}^{∞} \frac{(η_{0})_{n}(η_{1})_{n}}{η_{2})_{n}n!} z^{n},

where (η)_{n} := ∏_{0}^{n} −1(η + j). Let

Ĝ(z) :=_{2} F_{1}(η_{0}, η_{1}, \frac{1}{2}, \frac{x^{2}}{x^{2}+y^{2}} ),

where

η_{j} = \frac{1}{2} − \frac{2}{κ} − (−1)^{j} \frac{sqrt8aκ + (κ − 4)^{2}}{2κ} .

Lemma 4. Let κ > 0 and z ∈ ℍ. Then the limit

Z(z) := lim_{t↑τ(z)} \frac{y|g′_{t}(z)|}{Im g_{t}(z)}

exists a.s. We have Z(z) = ∞ a.s. if κ ≥ 8 and Z(z) < ∞ a.s. if κ < 8. Moreover, for all z ∈ ℝ we have

E[Z(z)^{a}] = \{ \begin{matrix}Ĝ(z)/Ĝ(1) & a < 1 − \frac{κ}{8}, κ < 8 \\ ∞ & a ≥ 1 − \frac{κ}{8}, a > 0 \\ 0 & a < 0, κ ≥ 8. \end{matrix}

We will not discuss the proof here, but consider only the geometric implications. For z_{0} ∈ ℍ and t < τ(z_{0}), let ϕ : ℍ → 𝔻 be a conformal mapping with ϕ(g_{t}(z_{0}) = 0. Set r_{t} := d(z_{0}, ∂H_{t}). Now the Schwarz lemma applied to ϕ ∘ g_{t}(z_{0} + r_{t}z) and the Koebe \frac{1}{4} Theorem applied to g_{t}^{−1} ∘ ϕ^{−1} give

\frac{Im g_{t}(z)}{2|g′_{t}(z_{0})} ≤ r_{t} ≤ frac2Im g_{t}(z_{0})|g′_{t}(z_{0})|.

Since lim_{t↑τ(z_{0})} r_{t} = d(z_{0}, γ[0, ∞) ∪ ℝ), we have

\frac{y_{0}}{2Z(z_{0})} ≤ d(z_{0}, γ[0, ∞) ∪ ℝ) ≤ \frac{2y_{0}}{Z(z_{0})},  (1)

and so the lemma tells us that γ[0, ∞) is dense in ℍ iff κ ≥ 8, an important part of proving (iii) above.

The key point of the proof of (ii) is the following lemma, which gives that z ∈ ⋃_{t>0} K_{t} a.s. whenever z ∈ ℍ.

Lemma 5. If κ > 4 and z ∈ \overline{ℍ} \ \{0\} then P (τ(z) < ∞) = 1.

Lemma 4 and (1, in turn, show that a.s. z ∉ γ[0, ∞), whenever z ∈ ℍ, since Z(z) < ∞ a.s. if κ < 8. For z ∈ ℝ \ {0}, we need the following variant of Cardy’s formula.

Lemma 6. If κ ∈ (4, 8) and

X := inf([i, ∞) ∩ γ[0, ∞))

then for all s ≥ 1,

P (X ≥ s) = π^{\frac{1}{2}} (\frac{4}{s})^{1− \frac{4}{κ}} \frac{_{2}F_{1}(1 − \frac{4}{κ}, 2 − \frac{8}{κ}, 2 − \frac{4}{κ}, \frac{1}{s})}{Γ(2 − \frac{4}{κ})Γ(\frac{4}{κ} − \frac{1}{2})}.

By Lemma 6, then, 1 ∉ γ[0, ∞) a.s. and so the same follows for every z ∈ ℝ \ {0} by scale and reflection invariance. This completes the proof of (ii).

One can easily show that, if κ > 8, z_{0} ∈ \overline{ℍ} is not swallowed a.s., for if t_{0} is the swallowing time and 0 < δ < d(z_{0}, γ[0, t_{0}]), then every z ∈ B(z_{0}, δ) is swallowed with z_{0}, giving P (Z(z_{0}) < ∞) > 0 by (1), contradicting Lemma 4. Thus a.s. z_{0} ∈ γ[0, ∞).

To complete the proof of (iii), it will suffice to prove the following result (which allows us to see that γ[0, ∞) is a.s. closed), which is interesting in its own right.

Theorem 7. For κ ≠ 8 the SLE_{κ} trace γ(t) is transient a.s., i.e.

P ( \underline{lim}_{t→∞} |γ(t)| = ∞) = 1.

The main tools are two lemmas.

Lemma 8. If κ ≤ 4 and x ∈ ℝ \ \{0\} then x ∉ \overline{γ[0, ∞)} a.s.

Lemma 9. If κ > 4, κ ≠ 8, and t > 0 then a.s. there is some ϵ > 0 such that K_{t} ⊂ ℍ ∩ B(0, ϵ).

We do not discuss the first lemma here, it being rather technical (as is Lemma 6, through which one obtains the analogous result for κ ∈ (4, 8)).

Concerning the second, we note that it suffices to show there is t > 0 such that

δ := P (0 ∉ \overline{H}_{t}) > 0,  (2)

for by scale invariance δ is independent of t, and since these events are monotone in t, this gives

P (∀t > 0, 0 ∉ \overline{H}_{t}) = δ.

Blumenthal’s 0-1 Law then gives δ = 1.

For κ ∈ (4, 8), scale and reflection invariance and Lemma 6 show that for a < b < 0 there is positive probability that [a, b] is swallowed all at once, i.e. P ([a, b] ∩ γ[0, ∞)∅) > 0, but a.s. there is t such that [a, b] ⊂ K_{t}. When this happens, we have [a, b] ∩ \overline{H}_{t} = ∅. Let S be the set of limit points of g_{τ(1)}(z) as z → 0 in H_{τ(1)}. Then S ⊂ (−∞, ξ(τ(1))) and {g_{τ(1)}(γ(t + τ(1))) − ξ(τ(1))}_{t≥0} =^{d} {γ(t)}_{t≥0}, so by the above argument there is positive probability that g_{τ(1)}(γ[τ(1), ∞)) ∩ S = ∅ but there is t > τ(1) such that g_{τ(1)}(K_{t} \ K_{τ(1)}) ⊃ S. This implies (2) for κ ∈ (4, 8).

For κ > 8, we assume that for all t > 0 a.s. 0 ∈ \overline{H}_{t} in attempt to reach a contradiction. We have seen that A.s. z_{0} ∈ γ[0, ∞) so if t_{0} = τ(z_{0}) then z_{0} = γ(t_{0}). Let t > 0. Conditional on F_{t_{0}}, g_{t_{0}}(K_{t+t_{0}} \K_{t_{0}})−ξ(t_{0}) =^{d} K_{t}, so by our assumption a.s. ξ(t_{0}) ∈ \overline{g_{t_{0}}(H_{t+t_{0}})}. Mapping by g_{t_{0}}^{−1}, we have z_{0} ∈ \overline{H}_{t+t_{0}}, i.e. z_{0} ∈ ∂K_{t+t_{0}}. Thus for all z ∈ ℍ and every t > 0 a.s. z ∈ ∂K_{t+τ(z)}. Therefore, a.s.

Λ_{2}(∂K_{t}) = ∫_{z∈ℍ} 1_{z∈∂K_{t}} dΛ_{2} ≥ ∫_{z∈ℍ} 1_{t>τ(z)} dΛ_{2},

where Λ_{2} denotes area measure. Then by Fubini’s theorem we have Λ_{2}(∂K_{t}) > 0 with positive probability for large t. This contradicts Corollary 2, however, and so proves the lemma for κ > 8.

We turn now to the proof of transience.

For κ > 4, Lemma 9 provides positive, random r > 0 such that K_{1} ⊂ ℍ ∩ B(z, r). Let ϵ > 0, and choose a constant r_{0} = r_{0}(κ, ϵ) > 0 so that P (r < r_{0}) < ϵ. The scaling property of SLE shows that for all t > 0,

P (K_{t} ⊂ ℍ ∩ B(0, r_{0}\sqrt{t})) ≥ 1 − ϵ,

and so

P (∃t′ > t : |γ(t′)| < r_{0}\sqrt{t}) < ϵ.

Since ϵ > 0 is arbitrary this provides the transience for κ > 4.

For κ ∈ [0, 4], γ is a simple path, and so a.s. there are two limit points x_{0}, x_{1} for g_{1}(z) as z → 0 in H_{1}. Since g_{1}(γ[1, ∞)) − ξ(1) =^{d} γ[0, ∞), Lemma 8 shows that a.s. x_{0}, x_{1} ∉ \overline{g_{1}(γ[1, ∞))}. This means 0 ∉ \overline{γ[1, ∞)}, and we may proceed much as for κ > 4 above.

References

[1] Jones, Peter W. and Makarov, N. Density Properties of Harmonic Measure. Ann. of MAth. (2), 142(3):427-455, 1995.

[2] Koskela, P. and Rohde, S. Hausdorff Dimension and Mean Porosity. Math. Ann., 309(4):593-609, 1997.

[3] Rohde, S. and Schramm, O., Basic Properties of SLE (preprint).

8 The harmonic explorer and its convergence to SLE_{4}

after O. Schramm and S. Sheffield A summary written by Nam-Gyu Kang

Abstract

We give a summary of results in [1].

8.1 Introduction

The harmonic explorer (HE) in a simply connected subset D of the hexagonal faces in the planar honeycomb lattice is a random interface γ, separating the cluster of black hexagon B containing left boundary from the cluster of white hexagon W containing right boundary, generated in a way that the future color of an undetermined hexagon x hit by γ will be determined by the color of the first black or white hexagon hit by a simple random walk on the space of hexagons starting from the undetermined hexagon. Thus the probability that the color of x will be black is the (discrete) harmonic measure ω(x, B, D).

Let ℍ denote the upper half plane in the complex plane. Suppose that γ : [0, ∞) → \overline{ℍ} is a simple continuous curve with starting from the origin and staying in ℍ during the time (0, ∞). By the Riemann mapping theorem, for each t ∈ [0, ∞) there exists a unique conformal map g_{t} : ℍ \ γ[0, t] → ℍ that satisfies the hydrodynamic renormalization: lim_{z→∞} g_{t}(z) − z = 0. The limit

cap_{∞}(γ[0, t]) := lim_{x→∞} z(g_{t}(z) − z)/2

is real and monotone increasing in t. We call it the capacity of γ[0, t] in ℍ from infinity. Since cap_{∞}(γ[0, t]) is continuous in t, it is natural to reparameterize γ so that cap_{∞}(γ[0, t]) = t. Then g_{t} satisfies the Loewner differential equation

∂_{t}g_{t}(z) = \frac{2}{g_{t}(z) − W (t)}, g_{0}(z) = z,  (1)

where W (t) = g_{t}(γ(t)). The function W (t) is continuous in t, and is called the driving parameter for γ. On the other hand, for a one-dimensional Brownian motion W (t) with speed κ starting from the origin, the path γ(t) := lim_{z→0} g_{t}^{−1}(z + W (t)) is called by the chordal SLE_{κ} in ℍ.

For κ < 8 and z = x + iy ∈ ℍ, the choral SLE trace γ passes to the left of z with the probability

\frac{1}{2} + \frac{Γ(4/κ)}{\sqrt{π}Γ(4/κ − 1/2)} \frac{x}{y} _{2}F_{1}(\frac{1}{2}, \frac{4}{κ}; \frac{3}{2}, −\frac{x^{2}}{y^{2}} ),

where _{2}F_{1} is a hypergeometric function. When κ = 2, 8/3, 4 and 8, the above probability simplifies to

1 + \frac{1}{π} \frac{xy}{|z|^{2}} − \frac{1}{π} arg z, \frac{1}{2} + \frac{x}{|z|}, 1 − \frac{1}{π} arg z and \frac{1}{2},

respectively. See [2].

The harmonic measure (1/π) arg z = ω(z, (−∞, 0), ℍ) gives the probability that a two dimensional Brownian motion starting at z first exits ℍ in (−∞, 0). Let Z = Z(t, z) := g_{t}(z) − W (t). Due to the conformal invariance, the harmonic measure gives the probability that a two dimensional Brownian motion starting at z first exits ℍ\γ[0, t] either in (−∞, 0) or on the left hand side of γ[0, t]. By Itô calculus,

d log Z = \frac{(4 − κ)}{2Z^{2}} dt − \frac{\sqrt{κ}}{Z} dB_{t}.

When κ = 4, we have d arg Z = − Im(2Z^{−1}) dB_{t}. Hence, at a fixed time t, the martingale (1/π) arg Z(t, z) represents the probability that, conditioned on the SLE_{4} path up until time t, the point z will lie to the left of the path. A discrete version of this property holds for the harmonic explorer.

8.2 The driving process converges to BM

We introduce the precise combinatorial notation for HE. We denote by T G, V (T G) the triangular grid in the plane and its vertices respectively. Let 𝔇 denote the set of domains D ⊂ ℂ whose boundary is a simple closed path contained in the grid T G. For a pair (V, h) of a set V ⊂ V (T G) and a bounded real valued function h defined on V, we denote by \overline{h} the discrete harmonic extension of h which is determined uniquely by the properties that \overline{h} agrees with h in V and is bounded harmonic at every vertex in V (T G) \ V.

Suppose D ∈ 𝔇. Let V_{0} := V (T G) ∩ ∂D. Given the centers \overline{v}_{0} and \overline{v}_{end} of two distinct edges of the grid T G on ∂D, let A_{+} [respectively, A_{−}] be the positively [respectively, negatively] oriented arc of ∂D from \overline{v}_{0} to \overline{v}_{end}.

Define 𝔥_{0} : V_{0} → \{0, 1\}, 𝔥_{0} = 1_{V_{0}∩A_{+}}. Assume \overline{v}_{n} ∉ ∂D unless n = 0. Also assume that V_{n}, 𝔥_{𝔫} : V_{n} → {0, 1} are defined. Let T_{n+1} be the triangle of T G containing \overline{v}_{n} but not \overline{v}_{n−1} unless n = 0. If n = 0, then let T_{1} be the triangle of T G containing \overline{v}_{0}. Let v_{n+1} be the vertex of T_{n+1} which is not on the edge containing \overline{v}_{n}, and let V_{n+1} := V_{n} ∪ \{v_{n+1}\}. Let v_{n+1}^{+} [respectively, v_{n+1}^{−}] be the two midpoints of edges of T_{n+1} that lie on the positively [respectively, negatively] oriented arcs of ∂T_{n+1} from \overline{v}_{n} to v_{n+1}. Let p_{n+1} := \overline{𝔥}_{n}(v_{n+1}) and {X_{n}}_{n=1}^{∞} be a sequence of i.i.d. uniform random variables in [0, 1]. If X_{n+1} ≤ p_{n+1} let \overline{v}_{n+1} := v_{n+1}^{+}, and otherwise let \overline{v}_{n+1} := v_{n+1}^{−}. Let denote c_{n} the center of T_{n}. Define the next step of the HE consist of the segments [\overline{v}_{n}, c_{n+1}] and [c_{n+1}, \overline{v}_{n+1}]. Also, let 𝔥_{n+1} agree with 𝔥_{n}, where 𝔥_{n} is defined and set 𝔥_{n+1}(v_{n+1}) := 1_{X_{n+1}≤p_{n+1}} if v_{n+1} ∉ V_{n}. This procedure a.s. terminates when \overline{v}_{N} = \overline{v}_{end} for some N and N is called the termination time. The HE is defined to be a random simple path from \overline{v}_{0} to \overline{v}_{N} in D.

For D ∈ 𝔇, let γ : [0, N] → D ∪ \{\overline{v}_{0}, \overline{v}_{N}\} be the HE path with γ(n) = \overline{v}_{n} for n = 0, 1, ⋯, N. There is a conformal map ϕ : D → ℍ onto ℍ such that that ϕ(\overline{v}_{0}) = 0, ϕ(\overline{v}_{N}) = ∞. Let p_{0} := ϕ^{−1}(i). For z ∈ ℂ, let rad_{z}(D) := inf{|w − z| : w ∉ D}. Instead of letting the mesh tend to 0, we take ρ = ρ(D, ϕ) := rad_{p_{0}}(D) → ∞. Let γ^{ϕ} be the path ϕ ∘ γ, parameterized by capacity from ∞ in \overline{ℍ}. Let W = W (t) denote the Loewner driving process for γ^{ϕ}. For n ∈ [0, N] let t_{n} := cap_{∞}(ϕ ∘ γ[0, n]), D̃_{n} := D \ γ[0, n], and let ϕ_{n} : D̃_{n} → ℍ be the conformal map normalized by ϕ_{n} ∘ ϕ^{−1}(z) − z → 0 as z → ∞ in ℍ. For fixed n ∈ ℕ, let A_{1} = A_{1}(n) be the event {n < N} and set p_{n} := ϕ_{n}^{−1}(i + W (t_{n})). For fixed R > 0, let A_{2} = A_{2}(n, R) be the event {rad_{p_{n}}(D) ≥ R}.

Proposition 1. For every δ ∈ (0, 1) there is an R = R(δ) > 0 such that the following holds. Fix any n ∈ ℕ. On the event A_{1}, let m be the least integer larger than n such that max{t_{m} − t_{n}, (W (t_{m}) − W (t_{n}))^{2}} ≥ δ^{2}. Then

𝔼[W (t_{m}) | γ[0, n]] = W (t_{n}) + O(δ^{3}),  (2)

and

𝔼[(W (t_{m}) − W (t_{n}))^{2} | γ[0, n]] = 4𝔼[t_{m} − t_{n} | γ[0, n]] + O(δ^{3})  (3)

hold on the event A_{1} ∩ A_{2}.

Let h_{n} = \overline{𝔥}_{n}. For each v ∈ V (T G) ∩ D, h_{n}(v) is a martingale. It implies

𝔼[h_{m}(v) − h_{n}(v)|γ[0, n]] = 0.  (4)

Since h_{j}(v) is discrete-harmonic, it is approximated by h̃(ϕ_{j}(v) − W (t_{j})) for a large D̃_{j}, where h̃(z) = 1 − (1/π) arg(z). The difference h_{m}(v) − h_{n}(v) can then be approximated by a function of t_{m} −t_{n} and W (t_{m})−W (t_{n}). Applying (4) for two distinct choices of v then gives the relations (2) and (3), which makes it possible to calculate the drift and diffusion of the Loewner driving process for the path traced by the HE.

Let z_{t} := g_{t} ∘ ϕ(w_{0}) = x_{t} + iy_{t}. Consider F (x, y, w) := h̃(z − w). We use a Taylor expansion of F about (x_{t_{n}}, y_{t_{n}}, W (t_{n})) to get an approximation of F (x_{t_{m}}, y_{t_{m}}, W (t_{m})) up to O(δ^{3}) terms. Let Δz = z_{t_{m}} − z_{t_{n}} = Δx + iΔy, ΔW = W (t_{m}) − W (t_{n}), Δh̃ = h̃(z_{t_{m}} − W (t_{m})) − h̃(z_{t_{n}} − W (t_{n})). Then

Δh̃ = ∂_{x}F Δx + ∂_{y}F Δy + ∂_{w}F ΔW + \frac{1}{2}∂_{w}^{2}F (ΔW)^{2} + O(δ^{3}).

On the other hand,

∂_{x}F = −∂_{w}F = −\frac{1}{π} Im \frac{1}{z − w}, ∂_{y}F = −\frac{1}{π} Re \frac{1}{z − w}, ∂_{w}^{2}F = \frac{1}{π} Im \frac{1}{(z − w)^{2}} .

By Loewner differential equation (1), Δz = 2Δt/(z_{t_{n}} − W (t_{n})) + O(δ^{3}). This implies

∂_{x}F Δx + ∂_{y}Δy = −\frac{4Δt}{π} Re \frac{1}{z_{t_{n}} − W (t_{n})} Im \frac{1}{z_{t_{n}} − W (t_{n})} + O(δ^{3})

= −\frac{2Δt}{π} Im \frac{1}{(z_{t_{n}} − W (t_{n}))^{2}} + O(δ^{3}).

Since 𝔼[Δh̃ | γ[0, n]] = O(δ^{3}), we have

O(δ^{3}) = 2 Im ((z_{t_{n}} − W (t_{n}))^{−2}) 𝔼[Δt | γ[0, n]]
− Im ((z_{t_{n}} − W (t_{n}))^{−1}) 𝔼[ΔW | γ[0, n]]
−\frac{1}{2} Im ((z_{t_{n}} − W (t_{n}))^{−2}) 𝔼[(ΔW)^{2} | γ[0, n]].

The proposition 1 and Skorokhod Embedding Theorem implies the following corollary.

Corollary 2. Fix T ≥ 1. As ρ → ∞, the restriction of t ↦ W (t/4) to [0, T] converges in law to the corresponding restriction of standard Brownian motion.

8.3 Local Hausdorff convergence to SLE_{4}

Let γ̃ be the SLE_{4} path in \overline{ℍ}. Also, let d_{H}(·, ·) denote the Hausdorff distance; that is, for two nonempty sets A, B ⊂ ℂ,

d_{H}(A, B) := max ( sup inf_{a∈A b∈B} |a − b|, sup inf_{b∈B a∈A} |a − b|).

Lemma 3. For every T ≥ 1 and ϵ > 0 there is some R = R(ϵ, T) such that if ρ > R, then there is a coupling of γ^{ϕ} and γ̃ such that

ℙ[ sup_{t∈[0,T]} d_{H}(γ̃[0, t], γ^{ϕ}[0, t]) > ϵ] < ϵ.

For a sequence (D, ϕ) of pairs such that ρ → ∞, we take a sequence of couplings of the corresponding γ^{ϕ}[0, T] and B such that sup{|W (t)−B(4t)| : t ∈ [0, T]} → 0 in probability as ρ → ∞. For fixed t ∈ [0, T], we get a coupling of B and a Hausdorff limit Γ_{t} of γ^{ϕ}[0, T] by passing to a subsequence. Note that Γ_{t} is contained in γ̃[0, T] and it is clearly connected. By the Carathéodory kernel theorem, the maps g_{t} : ℍ \ γ^{ϕ}[0, t] → H converge to the normalized conformal map from ℍ \ Γ_{t} to ℍ. Consequently, the capacity of Γ_{t} is t a.s. Thus, we conclude that Γ_{t} = γ̃[0, t] a.s. Since the limit does not depend on the subsequence, it follows that γ̃[0, t] is a.s. the Hausdorff limit Γ_{t} of γ^{ϕ}[0, t]. As t ∈ [0, T] is arbitrary, we conclude that a.s. Γ_{t} = γ̃[0, t] for every rational t ∈ [0, T]. The lemma now follows, since γ̃ is continuous and Γ_{t} is increasing.

8.4 Convergence in a strong topology

Proposition 4. For every fixed T > 0, there is a coupling of γ^{ϕ} and γ̃ such that

sup_{0≤t≤T} |γ^{ϕ}(t) − γ̃(t)| → 0

in probability as ρ → ∞.

For a sequence of pairs (D, ρ), with ρ → ∞, use Lemma 3 to couple γ̃ and the sequence γ^{ϕ} so that for each t > 0 the set γ̃[0, t] is a.s. the Hausdorff limit of γ^{ϕ}[0, t]. The proposition follows from the fact that the curves γ^{ϕ}[t_{1}, t_{2}] converge to γ̃[t_{1}, t_{2}] in the Hausdorff sense for all rational pairs 0 ≤ t_{1}, t_{2} ≤ T. It can be shown by proving that for all rationals t_{0} < t_{1} < t_{2} < t_{3} < T, the Hausdorff limit of γ^{ϕ}[t_{1}, t_{2}] is a.s. disjoint from both γ̃[0, t_{0}] and γ̃[t_{3}, T].

Since ϕ(\overline{v}_{end}) = ∞, γ^{ϕ} is transient; that is, lim_{t→∞} |γ^{ϕ}(t)| = ∞. The following is a uniform version of this statement.

Proposition 5. For every ϵ > 0 and R > 0 there is a T = T (R, ϵ) such that

ℙ[γ^{ϕ}[T, ∞) ∩ B(0, R) ≠ ∅] < ϵ

if ρ is sufficiently large.

To prove the uniform transience of γ^{ϕ}, the following property about the excursion measures is needed. For each R > 0, suppose that at each vertex v ∈ V (T G) ∩ ϕ^{−1}(\overline{ℍ} ∩ \overline{B(0, R)}) a simple random walk X^{v} is started, and the walk is stopped at the first time t ≥ 1 such that X^{v}(t) exits D or |ϕ(X^{v}(t))| ∉ (R, 2R). Then the expected number of walks which stop when |ϕ(X^{v}(t))| ≥ 2R is O(1).

Let d_{∗}(·, ·) be the metric on \overline{ℍ} ∪ \{∞\} given by d_{∗}(z, w) = |Ψ(z) − Ψ(w)|, where Ψ(z) := (z − i)/(z + i) maps \overline{ℍ} ∪ \{∞\} onto the closed unit disk.

Theorem 6. As ρ → ∞, the law of γ^{ϕ} tends to the law of the SLE_{4} path γ̃, with respect to uniform convergence in the metric d_{∗}. In other words, for every ϵ > 0 there is some R = R(ϵ) such that if ρ > R, then there is a coupling of γ^{ϕ} and γ̃ such that

ℙ[ sup_{t∈(0,∞)} d_{∗}(γ̃(t), γ^{ϕ}(t)) > ϵ] < ϵ.

The main theorem follows from proposition 4 and proposition 5.

References

[1] O. Schramm and S. Sheffield. The harmonic explorer and its convergence to SLE(4). Preprint, math.PR/0310210.

[2] O. Schramm. A percolation formula. Electron. Comm. Probab., vol. 6, paper no. 12, 115–120, 2001.

Nam-Gyu Kang, Massachusetts Institute of Technology email: kang@math.mit.edu

9 Hausdorff dimensions for SLE_{6}

after Vincent Beffara [1] A summary written by Tai Melcher

Abstract

Using the locality property and the chordal/radial equivalence of SLE when κ = 6, this paper proves that the dimension of the SLE_{6} trace is almost surely 7/4. This method is then adapted to prove that the dimension of the SLE_{6} boundary is almost surely 4/3, which was previously proved in [3] via the relation of SLE_{6} to planar Brownian motion.

9.1 Introduction

For all z ∈ \overline{ℍ}, consider the random collection of conformal maps g_{t} = g_{t}(z) defined by the initial value problem

ġ_{t}(z) = \frac{2}{g_{t}(z) − \sqrt{6}B_{t}} ,  (1)

with g_{0}(z) = z, where B_{t} is standard Brownian motion on ℝ. The solution to this equation exists up to the first time τ_{z} such that lim_{t↑τ_{z}} g_{t}(z) = \sqrt{6}B_{t}, and thus has simply connected domain H_{t} = {z ∈ ℍ : t < τ_{z}}. The set K_{t} = ℍ\H_{t} is a relatively compact subset of ℍ, and (K_{t})_{t≥0} is an increasing family such that ⋃ K_{t} is unbounded; thus, we say that K_{t} is growing towards ∞. There almost surely exists a (non-simple) curve γ : [0, ∞) → \overline{ℍ} that generates K_{t}, in the sense that H_{t} is the unbounded connected component of ℍ\γ([0, t]). We call this random collection of conformal maps g_{t} chordal SLE_{6}, (K_{t})_{t≥0} the SLE hull, and γ the SLE trace. It is known [6] that SLE_{6} is the scaling limit of critical percolation interfaces on the triangular lattice.

This paper studies certain geometric properties of SLE_{6}, in particular, the Hausdorff dimension of the trace γ([0, ∞)) and of the boundary ∂K_{t}. Informally speaking, we say that a set X has Hausdorff dimension d if N(r) is the number of balls of radius at most r required to cover X and N(r) ∼ 1/r^{d} as r ↓ 0. Formally, we define

H_{r}^{d}(X) := inf { ∑_{k=1}^{∞} diam(A_{k})^{d} : {A_{k}} is a cover of X, diam(A_{k}) ≤ r ∀k}.

Then the Hausdorff d-measure H^{d} and the Hausdorff dimension dim_{H} of X are given by

H^{d}(X) := lim_{r↓0} H_{r}^{d}(X) and dim_{H}(X) := inf{d ≥ 0 : H^{d}(X) = 0}.

Theorem 1. Almost surely, the Hausdorff dimension of γ([0, ∞)) is 7/4.

Theorem 2. Almost surely, the Hausdorff dimension of ∂K_{t} is 4/3.

The dimension of the SLE_{6} boundary was first determined in [3] by proving an equivalent statement for the exterior boundary of a Brownian path. The proofs in the present paper take advantage of features particular to SLE_{6}, namely, the locality property and the chordal/radial equivalence, to provide a more direct proof of Theorem 2 and the new result in Theorem 1. In [2], the author is able to extend his method to prove generally that, for all κ > 0,

dim_{H}(γ([0, ∞)) = 2 ∧ (1 + \frac{κ}{8}) almost surely.

Here we briefly state the properties particular to the κ = 6 case.

9.1.1 Locality and chordal/radial equivalence

The “locality” property of SLE_{6} can be stated generally as follows: the process is not affected by the boundary of the domain as long as it does not hit it. More precisely, let A ⊂ ℍ be a bounded hull such that 0 ∉ A, and let Φ_{A} denote the unique conformal transformation Φ of ℍ\A onto ℍ fixing 0 and ∞ with Φ(z)/z → 1 as z → ∞. Then (Φ_{A}(K_{t}))_{t≥0} is a continuously growing family of subsets of ℍ. Let

T := inf{t : K_{t} ∩ A ≠ ∅} and T′ := inf{t : K_{t} ∩ Φ_{A}(∂A) ≠ ∅}.

Then (Φ_{A}(K_{t}))_{t<T} has the same law as a time change of (K_{t})_{t<T′}.

We may use the locality property for SLE_{6} to study the conformal image of chordal SLE on certain subsets of the plane, for example, on the unit disk 𝕌. In particular, let Φ : 𝕌 → ℍ be a conformal homeomorphism. Then g_{t} ∘Φ is a solution to (1) with g_{0} ∘ Φ(z) = Φ(z), and we call this process a chordal SLE on the unit disk. If Φ(−1) = ∞, then (Φ^{−1}(K_{t}))_{t≥0} is an increasing family of subsets of 𝕌 growing towards −1. We will denote a chordal SLE on 𝕌 also by (K_{t})_{t≥0}.

The radial SLE process describes an increasing family of compact sets (K̃_{t})_{t≥0} of the closed unit disk that are growing from the boundary point 1 to 0. For all z ∈ \overline{𝕌}, let g̃_{t} = g̃_{t}(z) be defined by the initial value problem

∂_{t}g̃_{t}(z) = g̃_{t}(z)\frac{g̃_{t}(z) + β_{t}}{g̃_{t}(z) − β_{t}} , g_{0}(z) = z,

where β_{t} = e^{i\sqrt{6}B_{t}} is a time-scaled Brownian motion on ∂𝕌. This solution is well-defined up to the first time τ̃_{z} such that lim_{t↑τ̃_{z}} g̃_{t}(z) = β_{t}.

The main distinction between chordal and radial SLE in 𝕌 is that chordal SLE grows toward a boundary point (in this case -1) and radial SLE grows toward the interior point 0. Let

T := sup{t : −1 ∉ K_{t}} and T̃ := sup{t : 0 ∉ K̃_{t}}.

Then, up to a time change, the laws of (K_{t})_{t<T} and (K̃)_{t<T̃} are the same. Note that T is the first time when K_{t} disconnects 0 from −1, and similarly for T̃ and K̃_{t}. See [4, 5] for the original proofs of the above.

9.1.2 The construction

To prove the results on Hausdorff dimension, we will need the following general results. Let λ denote Lebesgue measure on the cube [0, 1]^{2}. Suppose that (C_{ε})_{ε>0} is a family of random Borelian subsets of [0, 1]^{2} such that C_{ε} ⊂ C_{ε′} whenever ε < ε′. Define the following conditions:

(1) There exist c_{−}, c_{+} > 0 such that, for all x ∈ [0, 1]^{2},

c_{−}ε^{s} ≤ P (x ∈ C_{ε}) ≤ c_{+}ε^{s}.

(2) There exists c > 0 such that, for all x ∈ [0, 1]^{2} and ε > 0,

P (λ(C_{ε} ∩ B(x, ε)) > cε^{2}|x ∈ C_{ε}) ≥ c.

(3) There exists c > 0 such that, for all x, y ∈ [0, 1]^{2} and ε > 0,

P ({x, y} ⊂ C_{ε}) ≤ cε^{2s}|x − y|^{−s}.

Proposition 3. Let C := ⋂ C_{ε}.

(i) If (1) and (2) hold, then a.s. dim_{H}(C) ≤ 2 − s. (ii) If (1) and (3) hold, then with positive probability dim_{H}(C) ≥ 2 − s.

Moreover, we have the following zero-one laws for the SLE trace and boundary (Lemmas 3 and 4 of the paper).

Lemma 4. For all d ∈ [0, 2],

P (dim_{H}(γ([0, ∞)) = d) ∈ {0, 1} and P (dim_{H}(∂K_{1}) = d) ∈ {0, 1}.

Thus, to verify Theorem 1, we must define appropriate random sets C_{ε} which satisfy conditions (1)-(3) with s = 1/4 such that ⋂ C_{ε} = γ([0, ∞)) (actually here we will define C_{ε} so that ⋂ C_{ε} ⊂ γ([0, ∞))) and apply the zero-one law to conclude, and similarly for Theorem 2 with s = 2/3.

9.2 Dimension of the trace

Let C be the intersection of the trace with the square [−1, 1] × [1, 3]. For ε > 0, define

C_{ε} = {z ∈ [−1, 1] × [1, 3] : d(z, C) ≤ ε}.

Note that C ⊂ γ([0, ∞)). Thus, proving the bounds in Proposition 3 for C proves the bounds for γ([0, ∞)). We must verify that the C_{ε} satisfy (1)-(3).

Let z ∈ [−1, 1] × [1, 3] and ε > 0 be such that B(z, ε) ⊂ [−1, 1] × [1, 3], and define the hitting time

T_{B(z,ε)} := inf{t : K_{t} ∩ B(z, ε) ≠ ∅}.

Then,

z ∈ C_{ε} ⟺ B(z, ε) ⊄ K_{T_{B(z,ε)}}.

The latter event is called nondisconnection; that is, K_{T_{B(z,ε)}} does not disconnect z from ∞. To prove estimates for nondisconnection probabilities for SLE_{6} in ℍ, we first consider similar probabilities for SLE_{6} in 𝕌.

Proposition 2 of the paper addresses the disconnection probabilities of chordal SLE_{6} (K_{t}) in 𝕌 for disks centered at 0 via the chordal/radial equivalence. For the hull (K̃_{t}) of a radial SLE_{6}, let

T̃_{B(z,r)} := inf{t : K̃_{t} ∩ B(z, r) ≠ ∅}.

Diffusion estimates from Section 1.2 of the paper give estimates for the probability that K̃T̃_{B(0,r)} does not disconnect B(0, r) from -1, thus estimating the probability that K_{T_{B(0,r)}} does not disconnect B(0, r) from -1.

For any B(z, r) ⊂ 𝕌, there exists a unique fractional linear transformation mapping 𝕌 to 𝕌 which fixes 1 and maps B(z, r) to a disk centered at 0. Corollary 1 then uses this fact and the previous estimates for disks centered at 0 to yield analogous bounds for the probability that K_{T_{B(z,r)}} does not disconnect B(z, r) from -1. A subsequent mapping to the upper-half plane then allows to us to apply Corollary 1 to nondisconnection probabilities for chordal SLE_{6} in ℍ, and thus verify condition (1) for C_{ε}.

Condition (2) now follows: for z ∈ C_{ε}, compactness implies that there exists z′ ∈ C such that |z − z′| = ε and B((z + z′)/2, ε/2) ⊂ C_{ε} ∩ B(z, ε), so

λ(C_{ε} ∩ B(z, ε)) ≤ λ(B((z + z′)/2, ε/2) = πε^{2}/4.

To verify condition (3), let z, z′ ∈ [−1, 1] × [1, 3] and ε < |z − z′|/2, and recall that the discrete exploration process γ_{δ} of critical percolation on a triangular lattice with mesh δ converges to the SLE_{6} trace as δ ↓ 0. Thus,

P (γ hits B(z, ε) and B(z′, ε)) = lim_{δ↓0} P (γ_{δ} hits B(z, ε) and B(z′, ε)).

Let A_{z}(r, R) denote the annulus centered at z with inner radius r and outer radius R. Then the event that γ_{δ} intersects both B(z, ε) and B(z′, ε) implies that there exist pairs of paths of open vertices and paths of closed vertices, each pair connecting

(i) the inner and outer boundaries of A_{z}(ε, |z − z′|/2) inside A_{z}(ε, |z − z′|/2), (ii) the inner and outer boundaries of A_{z′}(ε, |z−z′|/2) inside A_{z′}(ε, |z−z′|/2), (iii) ∂B((z + z′)/2, |z − z′|) to the real axis outside B((z + z′)/2, |z − z′|). These events are independent, and, by Corollary 2 of the paper, the probability of such crossings on the annulus A_{z}(r, R) may be estimated by (r/R)^{1/4}.

9.3 Dimension of the boundary

Let B = ∂K_{1}, and, for ε > 0, define

B_{ε} = {z ∉ K_{1} : d(z, K_{1}) < ε}.

Clearly, ⋂ B_{ε} = B. The proof that the random sets B_{ε} satisfy conditions (1)-(3) is analogous to the proof for the trace. Define the hitting time

T (z, ε) := inf{t : d(z, K_{t}) < ε}.

Then

z ∈ B_{ε} ⟺ L(z, ε) < ∞,

where L(z, ε) is the extremal distance between B(z, ε) and ∂𝕌 in 𝕌\K_{T (z,ε)}. Again, the right hand side is equivalent to a nondisconnection event, in particular, of z and ∂𝕌 by the SLE from time T (z, ε) to 1. Proving that this probability has the correct bounds again relies on the chordal/radial equivalence and diffusion estimates from Section 1.2 of the paper. Proposition 4 gives the estimate for chordal SLE_{6} on 𝕌, which can then be applied to chordal SLE_{6} on ℍ via a conformal mapping.

Condition (3) is determined via approximation by the discrete exploration process, as it was for the trace. Here, the intersection of a disk with the boundary of the exploration process indicates the existence of a pair of open and closed paths which cross the disk and a third disjoint path which prevents the process from closing a loop around the disk, keeping the disk out of the interior. Corollary 3 of the paper asserts that the probabilities of these crossings on an annulus A_{z}(r, R) may be estimated by (r/R)^{2/3}.

References

[1] Beffara, V., Hausdorff dimension of SLE_{6}. Ann. Probab. 32 no. 3B, 2606-2629, 2004.

[2] Beffara, V., The dimension of the SLE curves. Preprint, available at arxiv.org/abs/math/0211322.

[3] Lawler, G. F., Schramm, O., and Werner, W., The dimension of the Brownian frontier is 4/3. Math. Rev. Lett. 8 13-24, 2001.

[4] ————–, Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187 237-273, 2001.

[5] ————–, Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187 275-308, 2001.

[6] Smirnov, S., Critical percolation in the plane. C. R. Acad. Sci. Paris Seŕ. I Math. 333 239-244, 2001.

Tai Melcher, Berkeley email: melcher@stat.berkeley.edu

10 Basic Properties of SLE

after S.Rhode and O.Schramm A summary written by Brady Max NewDelman

Abstract

We prove that for all κ ≠ 8, the SLE trace is a continuous path a.s. This is Theorem 5.1.

10.1 The SLE Trace

Let B_{t} be Brownian motion on ℝ started from B_{0} = 0. For κ ≥ 0 let ξ(t) := \sqrt{κ}B_{t} and for each z ∈ \overline{ℍ} \ \{0\} let g_{t}(z) be the solution of the ordinary differential equation

∂_{t}g_{t}(z) = \frac{2}{g_{t}(z) − ξ(t)},  g_{0}(z) = z.  (1)

When ξ(t) is a continuous function, (1) is called the Chordal Loewner Equation with driving function ξ(t). If we let τ(z) be the first time τ such that 0 is a limit point of g_{t}(z) − ξ(t) as t ↗ τ, then we define

H_{t} := \{z ∈ ℍ : τ(z) > t\}, K_{t} := \{z ∈ \overline{ℍ} : τ(z) ≤ t\}.

It will be shown that the solution to (1) is a conformal homeomorphism g_{t}(z) : H_{t} → ℍ. Let f_{t}(z) := g_{t}^{−1}(z), ĝ_{t}(z) := g_{t}(z) − ξ(t), and f̂_{t}(z) := f_{t}(z + ξ(t)), then we can define the trace γ of the SLE by

γ(t) := lim_{ℍ∋z→0} f̂_{t}(z).

Theorem 5.1 assures us that this limit exists and is a continuous path a.s. for κ ≠ 8

10.2 Preliminaries

There are two results and two formulas that will be used throughout the paper.

10.2.1 Results

1. H_{t} = {z ∈ ℍ : σ(z) > t} where σ = inf{t : ℑ(g_{t}(z)) = 0}. This follows from the identity

ℑ(∂_{t}g_{t}(z)) = \frac{−2ℑ(g_{t}(z))}{|g_{t}(z) − ξ(t)|^{2}} ,

which also show that ℑ(g_{t}(z)) is monotonically decreasing.

2. H_{t} is unbounded a.s.

Proof: Let R_{t} = max of \sqrt{t} and sup\{|ξ(t)| : 0 ≤ s ≤ t\} and let σ(z) be the first time |g_{s}(z) − z| ≥ R_{t}. If |z| > 4R_{t} and s < t ∧ σ, then |∂_{s}g_{s}(z)| ≤ 1/R_{t}. Hence |z − g_{s}(z)| ≤ s/R_{t}; 0 ≤ s ≤ t ∧ σ. Therefore either σ > t or σ ≥ R_{t}^{2}. But R_{t}^{2} ≥ t, so σ ≥ t. So we must have {z ∈ ℍ : |z| > 4R_{t}} ⊂ H_{t} and H_{t} is unbounded.

10.2.2 Formulas

1.

∂_{t} log |g′_{t}(z)| = ℜ (\frac{∂_{z}∂_{t}g_{t}(z)}{g′_{t}(z)}) = ℜ (g′_{t}(z)^{−1}∂_{z} \frac{2}{g_{t}(z) − ξ(t)})
= −2ℜ((g_{t}(z) − ξ(t))^{−2}).

2. Set u = u(z, t) := log ℑ(g_{t}(z)), then

∂_{t}u = −2|g_{t}(z) − ξ(t)|^{−2},

and 1) gives

∂_{u} log |g′_{t}(z)| = \frac{ℜ((g_{t}(z) − ξ(t))^{2})}{|g_{t}(z) − ξ(t)|^{2}}.

Throughout this paper we assume the speed κ ≠ 8. The proof of Theorem 5.1 depends on chapters 3 and 4. In Chapter 3, we derive bounds on quantities related to 𝔼[|g′_{t}(z)|^{a}], and for the moments of |f̂′_{t}|. In Chapter 4, we reduce the problem of studying the set of limit points of γ(t) := lim_{ℍ∋z→0} f̂_{t}(z) to lim_{y↘0} f̂_{t}(iy).

10.3 Summary of Chapter 3

We want to study the moments of both |f̂′_{t}| and |g′_{t}|. It may appear that f̂_{t} is easier to deal with since it has domain the upper half plane; however, f_{t} is given by the PDE,

∂_{t}f_{t}(z) = −∂_{z}f_{t}(z)\frac{2}{z − ξ(t)}.

The problem is easily rectified when we consider a two-sided Brownian motion. Then we have:

Lemma 1. For all fixed t ∈ ℝ the map z ↦ g_{t}(z) has the same distribution as the map z ↦ f̂_{t}(z) − ξ(t).

From Lemma 1 we see that the two functions f_{t} and g_{t} are more or less interchangeable. The first result of chapter 3 is the following:

Theorem 2. Let z = x + iy with y ≠ 1. Let ν = sign log y, F_{t} = σ(ξ_{vs} : s ∈ [0, t]), and let α,β,γ ∈ ℝ be the numbers satisfying

α = \frac{νκβ^{2}}{2} ,

γ = 2β + \frac{νκβ}{2} (β − 1),

if we define the stopping time T := inf{t ∈ ℝ; ℑg_{t}(z) = 1}, then 𝔼[|ĝ_{T} (z)|^{2β}|ĝ′_{T} (z)|^{γ}] = y^{a}(x^{2} + y^{2})^{β}.

Outline of Proof. We let A_{t}(z) = |ℑ(ĝ_{vt}(z))|^{a}, B_{t}(z) = |ĝ′_{vt}(z)|^{2β}, C_{t}(z) = |ĝ′_{vt}(z)|^{γ} and we like to show that G_{t} = A_{t}B_{t}C_{t} is a local martingale for the filtration (F_{t})_{t≥0}, and this will be obtained if the Ito derivative of G_{t} does not have a drift term.

Taking derivatives we get from Ito’s formula

dG_{t} = \frac{−G_{t}2βx_{t}}{x_{t}^{2} + y_{t}^{2}} dξ(t) + G_{t} [\frac{−2να}{x_{t}^{2} + y_{t}^{2}} − 2νγ \frac{x_{t}^{2} − y_{t}^{2}}{(x_{t}^{2} + y_{t}^{2})^{2}} + 4κβ \frac{x_{t}^{2} − y_{t}^{2}}{(x_{t}^{2} + y_{t}^{2})^{2}}

+\frac{κβ}{x_{t}^{2} + y_{t}^{2}} + 2κβ(β − 1)\frac{x_{t}^{2}}{(x_{t}^{2} + y_{t}^{2})^{2}} ] dt,

where x_{t} = ℜ(ĝ_{vt}(z)) and y_{t} = ℑ(ĝ_{vt}(z)). The constants α, γ are properly chosen so

dG_{t} = \frac{−G_{t}2βx_{t}}{x_{t}^{2} + y_{t}^{2}} dξ(t)

and, hence G_{t} is a local martingale. Applying Doob’s optional stopping theorem with stopping time T, we get Theorem 2.

Changing constants, let ẑ = x̂ + iŷ and b ∈ ℝ. Assuming ŷ ≠ 1

Let ν := −sign log ŷ

a := 2b + \frac{νκb(1 − b)}{2}

λ := 4b + \frac{νκb(1 − 2b)}{2}

and set F (ẑ) = F_{b}(ẑ) := ŷ^{a}𝔼[|ĝ_{T} |^{2b}|g′_{T} (z)|^{a}] then Theorem 2 tells us

F_{b}(ẑ) = ŷ^{λ}(1 + x̂^{2}/ŷ^{2})^{b},

and this is the form that gets used in the following Corollary with gives us our desired bounds on the moments of |f̂′_{t}(x + iy)|.

Corollary 3. Let b ∈ [0, 1 + 4/κ], and define λ and a as above with ν = 1. There is a constant C(κ, b), depending only on κ and b such that the following estimate holds for all t ∈ [0, 1], y, δ ∈ [0, 1] and x ∈ ℝ.

ℙ[|f̂′_{t}(x + iy)| ≥ δ/y] ≤ C(κ, b)(1 + x^{2}/y^{2})^{b}(y/δ)^{λ}Θ(δ, a − λ),

where

Θ(δ, s) = \{ \begin{matrix}δ^{−s} : & s > 0 \\ 1 + | log δ| : & s = 0 \\ 1 : & s < 0. \end{matrix}

After the proof of Corollary 3 we get to the main result of chapter 3. We prove the continuity of β(t) := lim_{y↘0} f̂_{t}(iy) in the following:

Theorem 4. Define

H(y, t) := f̂_{t}(iy), y > 0, t ∈ [0, ∞).

If κ ≠ 8, then a.s. H(y, t) extends continuously to [0, ∞) × [0, ∞).

Outline of Proof. We partition [0, 1] × [0, 1] into rectangles R(j, k) := [2^{−j−1}, 2^{−j}] × [k2^{−2j}, (k + 1)2^{−2j}], where j, k ∈ ℕ with k < 2^{2j}. Let d(j, k) := diamH(R(j, k)), our goal is to show

∑_{j=0}^{∞} ∑_{k=0}^{2^{2j}−1} ℙ[d(j, k) > 2^{−jσ}] < ∞,  (2)

where a, λ are as in Theorem 2, and 0 < σ < (λ − 2)/(max{a, λ}), b = 8 + κ/(4κ).

The idea of the proof is to decompose the time interval [k2^{−2j}, (k + 1)2^{−2j}] into (random) sub-intervals using partition points t̂_{N}, t̂_{N−1}, …, t̂_{0} such that the change of ξ on each interval is controlled. Then it is not hard to control f̂_{t} on each sub-interval.

We then let S := {x + iy : |x| ≤ 2^{−j+3}, y ∈ [2^{−j−1}, 2^{−j+3}]} and derive

H(R(j, k)) ⊂ ⋃_{n=0}^{N} f̂t̂_{n}(S).

This gives us our first look at why we where trying to place bounds on |f̂′_{t}(z)|, since the diameter, d(j, k), of H(R(j, k)) is controlled by the distortion of f̂_{t} on S. We finish the proof by proving (2) and showing the limit of H(y, t) as (y, t) approaches the t-axis exists and is continuous.

10.4 Summary of Chapter 4

Having obtained β(t) := lim_{y↘0} f̂_{t}(iy) exists and is continuous, we want to show that β(t) = γ(t). This follows from Theorem 5.

Theorem 5. Let ξ : [0, ∞) ⟶ ℝ be continuous, and let g_{t} be the corresponding solution to (1). Assuming that β(t) exists for all t ∈ [0, ∞) and is continuous. Then g_{t}^{−1} extends continuously to \overline{ℍ} and H_{t} is the unbounded connected component of \overline{ℍ} \ β([0, t]), for every t ∈ [0, ∞).

The proof is based on the basic property of conformal maps. Suppose g : D ⟶ 𝕌 is a conformal homeomorphism. If α : [0, 1) → D is a path such that the limit l_{1} = lim_{t↗1} α(t) exists, and α̃ : [0, 1) → D is another such path with lim_{t↗1} α̃(t) exists and lim_{t↗1} g ∘ α(t) = lim_{t↗1} g ∘ α̃(t). Then lim_{t↗1} α(t) = lim_{t↗1} α̃(t) = l_{1}.

10.5 Theoem 5.1

We have now established all the results needed to show that the SLE trace is a continuous path a.s. By Theorem 4, a.s. lim_{y↘0} f̂_{t}(iy) exists for all t and is continuous. Therefore we can apply Theorem 5, and the theorem follows. Moreover we showed that H_{t} is the unbounded component of \overline{ℍ} \ γ([0, t]).

References

[1] S.Rhode, O.Schramm, Basic Properties of SLE, arXiv: math.PR /0106036.

[2] M. Biskup, Mulitifractal Spectrum of SLE Boundaries.(unpublished).

[3] G. Lawler,(2005) Conformally Invariant Processes in the Plane, American Mathematical Society.

Brady Max NewDelman, UCLA email: bnewdelm@math.ucla.edu

11 Critical percolation in the plane

After Stanislav Smirnov’s paper [11] a summary written by Gábor Pete

Abstract

We present Smirnov’s proof that critical site percolation on the triangular lattice satisfies Cardy’s conformally invariant crossing probability formulas in the scaling limit. Then we sketch how this result implies existence, uniqueness, and conformal invariance of the scaling limit, which limit can be described by the SLE_{6} process.

11.1 Introduction

Let T_{δ} be the triangular lattice with small mesh size δ, and color the vertices blue or yellow independently with probability p = 1/2. This is well known to be the critical value p_{c} here; see [5] for background on percolation. It was conjectured for a long time that critical percolation has a conformally invariant scaling limit, see e.g. [8], and conjectured by Schramm [10] that percolation interfaces in the limit are described by SLE_{6} curves. Defining exactly what the full scaling limit of percolation should be is not quite trivial, so the simplest approach is to say that certain macroscopic quantities are conformally invariant, and then this information might be enough to conclude that there is a unique conformally invariant full scaling limit. Such a macroscopic quantity is Cardy’s crossing probability formula [4] for the probability of a blue crossing between the opposite sides of a conformal rectangle Ω. The general formula is a hypergeometric function of the cross-ratio of the the four vertices of the rectangle, but it has a very simple form in an equilateral triangle:

Theorem 1 (Cardy’s formula in Carleson’s form). Consider an equilateral triangle abc with side length 1, and a point x ∈ [bc]. As δ → 0, the probability of a blue crossing on T_{δ} from [xb] to [ac] tends to |xb|.

This was proved by Smirnov [11], and our main goal in this note is to present this proof. Theorem 1 clearly follows from Theorem 6 below, which says that certain crossing probabilities H_{α}^{δ}(z), for z in a conformal triangle Ω, converge to a harmonic function h_{α}(z) on Ω, which changes conformally with the domain. Actually, there will be a “harmonic conjugate triple” h_{α}(z), α ∈ {1, τ, τ^{2}}, where τ = e^{2πi/3}. This three-fold symmetry is not a consequence of the symmetries of the triangular lattice T , but should be intrinsic to critical percolation. However, Smirnov’s proof works only for T_{δ}, while conformal invariance is conjectured to hold for any lattice. It is also interesting to note that the proof does not explicitly use that p = 1/2 is the critical probability; only the self-duality 1 − p = p matters.

Smirnov also sketched a proof that certain percolation interfaces converge to certain random curves given by SLE_{6}, and sketched a way to get a scaling limit for the entire collection of interfaces on the plane. These proofs were written down correctly in details only in a long recent paper by Camia and Newman [3], so we will just briefly sketch the main ideas in the final subsection.

Convergence to SLE_{6} makes it possible to compute several critical exponents for percolation on T , e.g. the one for the percolation probability: θ(p) = (p − p_{c})^{5/36+o(1)} as p → p_{c}+. An important ingredient here are Kesten’s scaling relations [6], which say that it is enough to compute the exponents for ℙ(A_{R}^{1}) and ℙ(A_{R}^{2}) at criticality, where A_{R}^{j} is the event of j disjoint clusters going from the vicinity of a point to distance R. These are tractable events for the SLE_{6}, and it was shown by [12] and [9] e.g. that ℙ(A_{R}^{1}) = R^{−5/48+o(1)}.

11.2 Elementary percolation bounds

Bond percolation on ℤ^{2} and site percolation on T , with parameter p = 1/2, have obvious self-duality properties, which imply, for example, that if R is an L × K rectangle (rhombus in the triangular setting), then the horizontal and vertical crossing probabilities add up to 1, i.e. ℙ(Hor(R)) + ℙ(V er(R)) = 1. Similarly, for any annulus A = B(r) \ B(s), where r > s > 0 and B(r) is the ball of radius r around the origin, the transversal and circular crossing probabilities satisfy ℙ(T rv(A)) + ℙ(Cir(A)) = 1. This duality is what makes planar percolation so special, and in fact tractable. In this section everything holds both for bond percolation on ℤ^{2} and site percolation on T , with parameter p = 1/2. We omit the proofs.

The following two lemmas are known as the Russo-Seymour-Welsh estimates. The simplest proof, involving one page of elementary planar mapdrawing and the Harris-FKG inequality, is by Bollobás and Riordan [2].

Lemma 2 (RSW for rectangles). For any ρ ∈ ℝ_{+} fixed, there are constants 0 < c_{1}(ρ) < c_{2}(ρ) < 1 such that for any ρL × L rectangle R, we have ℙ(Hor(R)) ∈ [c_{1}(ρ), c_{2}(ρ)] in critical percolation.

This easily implies the following version, which we will use several times.

Lemma 3 (RSW for annuli). For any ρ ∈ ℝ_{+} fixed, there are constants 0 < c′_{1}(ρ) < c′_{2}(ρ) < 1 such that for any annulus A = B(ρR) \ B(R), we have ℙ(T rv(A)) ∈ [c′_{1}(ρ), c′_{2}(ρ)] in critical percolation.

The following two lemmas are needed in proving that the scaling limits of certain percolation interfaces are well-behaved random curves. The first lemma is proved in Appendix A of [9], the second one is Lemma 5 of [7]. The proofs use the RSW estimates, the Harris-FKG and the v.d.Berg-Kesten inequalities, and a standard color-flipping trick, which we will see in the proof of Lemma 7. Note that once we know convergence to SLE_{6}, the exact exponents for the probabilities below can be explicitly computed [12].

Lemma 4 (Three arms in the half-plane). There exist 0 < ϵ and C < ∞ such that for all 0 < r < R, the probability that there exist two disjoint blue and a yellow transversal crossing in the half-annulus (B(R) \ B(r)) ∩ {z : ℑz > 0}, is at most C(r/R)^{1+ϵ}.

Lemma 5 (Six arms in the plane). There exist 0 < ϵ and C < ∞ such that for all 0 < r < R, the probability that there exist six transversal crossings, not all of the same color, in the annulus B(R) \ B(r), is at most C(r/R)^{2+ϵ}.

11.3 The proof of Cardy’s formula

Consider a simply connected domain Ω whose boundary ∂Ω is a smooth curve, and take three accessible boundary points, labeled counterclockwise a(α), a(τα), a(τ^{2}α). There are harmonic functions

h^{Ω}(a(α), a(τα), a(τ^{2}α), z) ≡ h_{α}(z), α ∈ {1, τ, τ^{2}},

which are the unique solutions of the following mixed Dirichlet-Neumann problem:

h_{α} = 1 at a(α), h_{α} = 0 on the arc a(τα)a(τ^{2}α),

\frac{∂}{∂τν} h_{α} = 0 on the arc a(α)a(τα),  (1)

\frac{∂}{∂(−τ^{2}ν)}h_{α} = 0 on the arc a(τ^{2}α)a(α),

where ν is the counter-clockwise-pointing unit tangent to ∂Ω. This problem is conformally invariant, and a domain in which the solution is particularly nice, is the equilateral triangle, with vertices a(α), a(τα), a(τ^{2}α). Here h_{α}(z) is just the properly scaled distance of z from the side a(τα)a(τ^{2}α).

Now define the event Q_{α}(z) for α ∈ {1, τ, τ^{2}} and z ∈ Ω, as the occurrence of a blue simple path in T_{δ} going from the arc a(α)a(τα) to the arc a(τ^{2}α)a(α) and separating z from the arc a(τα)a(τ^{2}α). Then we define the functions

H^{Ω,δ}(a(α), a(τα), a(τ^{2}α), z) ≡ H_{α}(z) := ℙ(Q_{α}(z)), α ∈ {1, τ, τ^{2}}.

Theorem 6 (Main convergence result). As δ → 0, the functions H_{α}^{δ} converge uniformly in Ω to the functions h_{α}.

Proof. Take β ∈ {1, τ, τ^{2}}. If z is the center of some lattice triangle, and z + η is the center of one of the adjacent triangles, denote by P_{β}(z, η) the probability of the event Q_{β}(z + η) \ Q_{β}(z). Then the discrete derivative of H_{β} can be written as

\frac{∂}{∂η} H_{β}(z) := H_{β}(z + η) − H_{β}(z) = P_{β}(z, η) − P_{β}(z + η, −η).  (2)

Lemma 7 (2π/3-Cauchy-Riemann equations). P_{β}(z, η) = P_{τβ}(z, τη).

Proof. Name the vertices of the lattice triangle of z by X, Y, Z, as shown on Figure 1. One can see that the event Q_{β}(z + η) \ Q_{β}(z) is equivalent to the occurrence of three disjoint simple paths, joining X, Y, Z to the arcs a(τβ)a(τ^{2}β), a(τ^{2}β)a(β), a(β)a(τβ), and colored yellow, blue, blue, respectively. There can of course be several choices for these paths; choose the counterclockwise-most yellow path from X and the clockwise-most blue path from Y.

But in this description, because of p = 1/2, we can change the color of the third arm to yellow, without changing the probability of the configuration.

Figure 1: Yellow arm from X, blue arms from Y and Z.

This new configuration, after switching the colors of all sites, is equivalent to the event Q_{τβ}(z + τη) \ Q_{τβ}(z), and we proved the lemma.

Lemma 8 (Hölder norm estimates). There are positive constants ϵ and C depending on the domain Ω only, such that H_{β} has ϵ-Hölder norm at most C. In particular,

P_{β}(z, η) ≤ C′δ^{ϵ}.  (3)

The boundary values of H_{β} are zero on the arc a(τβ)a(τ^{2}β) and tend to 1 at the point a(β) as δ → 0.

Proof. First note that H_{β}(z)−H_{β}(z′) = ℙ(Q_{β}(z)\Q_{β}(z′))−ℙ(Q_{β}(z′)\Q_{β}(z)), so it is enough to bound the two terms on the RHS. Moreover, if ∂Ω is nice enough, then can suppose that one of the boundary arcs, say a(α)a(τα), is at least some fixed positive distance from z and z′.

As in the proof of Lemma 7, for Q_{β}(z) \ Q_{β}(z′) to occur there must be a monochrome cluster connecting the segment [z, z′] to the boundary arc a(α)a(τα). Then can use Lemma 3 for a nested sequence of annuli of a fixed shape between [z, z′] and a(α)a(τα) to get the desired polynomial bound.

For the boundary values, again Lemma 3 can be used.

Similarly to (3), it can be proved that if |z − z′| = O(δ), then |P_{β}(z, η) − P_{β}(z′, η)| = O(δ^{1+ϵ}). This and Lemma 7 immediately give that

\frac{∂}{∂η} H_{β}(z) = \frac{∂}{∂(τη)}H_{τβ}(z) + O(δ^{1+ϵ}),

which means that the triple H_{1}, H_{τ}, H_{τ^{2}} satisfy the approximate discrete “Cauchy-Riemann equations for triples”, and hence is an approximate discrete harmonic conjugate triple. We are going to prove that as δ → 0, the functions H_{α} have limits which form an exact harmonic conjugate triple, with the boundary conditions specified in (1), and hence these limits must be the h_{α}. For this, technically the simplest is to use contour integrals instead of derivatives. Note, however, that Lemma 7 holds for any triangulated graph, while the following lemma about contour integrals is the ingredient of the proof that heavily uses the symmetries of T .

Take an equilateral triangular contour Γ with vertices in the centers of the lattice triangles, and with bottom side parallel to the real axis. Denote its vertices by x(1), x(τ), x(τ^{2}) counterclockwise, starting with the top one. For a function H(z) on triangles, we define the discrete contour integral

∫_{Γ}^{δ} H(z)dz := δ ∑_{z∈x(τ)x(τ^{2})} H(z) + δτ ∑_{z∈x(τ^{2})x(1)} H(z) + δτ^{2} ∑_{z∈x(1)x(τ)} H(z).

Lemma 9 (Contour integrals vanish). For any contour Γ as above, with total length ℓ, and any β ∈ {1, τ, τ^{2}}, we have

∫_{Γ}^{δ} H_{β}^{δ}(z)dz = ∫_{Γ}^{δ} \frac{1}{τ}H_{τβ}^{δ}(z)dz + O(ℓδ^{ϵ}).

Proof. Color all triangles in a chessboard fashion, so that the triangles with centers on Γ are black. Let B be the set of black centers on or inside Γ, while W be the set of white centers inside Γ. Fix α ∈ {1, τ, τ^{2}}, and take η := δ/\sqrt{3}e^{πi/6}(x(τ^{2}α) − x(τα)) and η′ := e^{πi/3η}. Using (2), (7) and (3), we get

∑_{z∈B\x(α)x(τ^{2}α)}(H_{β}(z + η) − H_{β}(z)) =  (4)

∑_{z∈B\x(α)x(τα)}(H_{τβ}(z + τη) − H_{τβ}(z)) + O(ℓδ^{ϵ−1}).

Note how the domain of summation is changed slightly, corresponding to the change in the direction of the translation. Similarly,

∑_{z∈W}(H_{β}(z + η′) − H_{β}(z)) = ∑_{z∈W}(H_{τβ}(z + τη′) − H_{τβ}(z)) + O(ℓδ^{ϵ−1}). (5)

Combine (4) and (5) and cancel the terms in the interior of Γ to get

∑_{z∈x(α)x(τ^{2}α)} H_{β}(z) − ∑_{z∈x(τα)x(τ^{2}α)} H_{β}(z) =  (6)

∑_{z∈x(α)x(τα)} H_{τβ}(z) − ∑_{z∈x(α)x(τ^{2}α)} H_{τβ}(z) + O(ℓδ^{ϵ−1}).

Now, −\frac{δ}{2}(6)|_{α=1} − i\frac{δ\sqrt{3}}{2}(6)|_{α=τ} + \frac{δ}{2}(6)|_{α=τ^{2}} gives the desired indentity.

From Lemma 8 we know that from any sequence of functions H_{α}^{δ}, where δ → 0, one can choose a subsequence converging uniformly in Ω. We need to prove that if for some subsequence δ_{j} → 0 the limit is some function f_{α}, then f_{α} ≡ h_{α}.

First of all, from Lemma 9 we get that for any Γ as above,

∫_{Γ} f_{β}(z)dz = ∫_{Γ} \frac{1}{τ}f_{τβ}(z)dz.  (7)

Fixing α ∈ \{1, τ, τ^{2}\}, and considering (7)|_{β=α} − (\frac{1}{2} + \frac{i}{2\sqrt{3}}) · (7)|_{β=τα}, we get

∫_{Γ} (f_{α}(z) + \frac{i}{\sqrt{3}}(f_{τα}(z) − f_{τ^{2}α}(z))) dz = 0.  (8)

By Morera’s theorem, the function in (8) is analytic, so for any α ∈ {1, τ, τ^{2}}, the function f_{α} is harmonic, with harmonic conjugate \frac{1}{\sqrt{3}}(f_{τα}(z) − f_{τ^{2}α}(z)). Hence f_{1}, f_{τ}, f_{τ^{2}} form a harmonic conjugate triple, and indeed, expanding \frac{∂}{∂(τx)} = −\frac{1}{2} \frac{∂}{∂x} + \frac{\sqrt{3}}{2} \frac{∂}{∂y}, and using the classical Cauchy-Riemann equations, one can easily show that for any unit vector η,

\frac{∂}{∂η} f_{α} = \frac{∂}{∂(τη)}f_{τα}.

From this and from f_{α}|_{a(τα)a(τ^{2}α)} = 0 we see that the f_{α}’s satisfy the Dirichlet-Neumann conditions (1). Since that problem has a unique solution, we must have f_{α} ≡ h_{α}, and the proof of Theorem 6 is complete.

11.4 Continuum scaling limit

We will consider two natural ways to define percolation interfaces. Restricting our attention to percolation in an equilateral triangle abc, first consider the unique vertex w on bc with neighbors accessible by a simple blue path from ca and a simple yellow path from ab. Let ω be the union of the blue path closest to c with the yellow path closest to a. Hence this ω is a path with two simple halves, touching bc at a unique point. Note that it follows from Theorem 1 that, as δ → 0, the distribution of w tends to the uniform on bc.

The second interface is constructed by fixing the color on ca to be blue and the color on ab and bc to be yellow, and then taking the unique path γ through the faces of T_{δ} from a to c which has all blue vertices on its left, and all yellow vertices on its right. This γ is called the exploration path from a to c. Note that the monochromatic sets along γ are not necessarily simple paths, but can have long tentacles, i.e., γ can “almost touch” itself.

Theorem 10 (Convergence to SLE_{6}). As δ → 0, the law of ω converges to the outer boundary of the filling of chordal SLE_{6} from a to c, stopped at hitting bc. The law of γ converges to this SLE_{6} itself. Note that by the locality property of SLE_{6}, the target point can also be b instead of c.

Very rough sketch of proof. It follows from the Hölder regularity estimates of [1] that, as δ → 0, there are subsequential limits for the laws of ω and γ. One needs to show that all these limits have certain properties that uniquely characterize SLE_{6} and its filling.

For ω, one first proves that the limiting curve consists of two simple path halves, separated by the unique touching point at bc. A simple path half becoming non-simple in the scaling limit would correspond to a multicolored six-arm event centered at the touching point. On T_{δ}, there are O(δ^{−2}) points inside abc, so, by Lemma 5, with probability tending to 1, there is no six-arm event in abc. Furthermore, a double touch of the curve at bc would correspond to a multicolored three-arm event at bc. This boundary arc has size O(δ^{−1}), so Lemma 4 implies that such a three-arm event does not happen. Then the complete distribution of the limiting curve can be determined using Cardy’s formula, and convergence to the outer boundary of SLE_{6} follows.

For γ, one has to first shows that there is a good time parametrization of the limiting curve, that satisfies the spatial Markov property. For this, one thing to check is that the curve never enters into a previously trapped area, i.e., that the discrete exploration process enters a fjord with vanishing width with a probability tending to 0. Such an event would imply eight arms around the entrance of the fjord, not all of the same color. But this event has a small enough probability due to Lemma 5, hence it does not happen anywhere inside abc. After this, the hitting probabilities given by Cardy’s formula prove convergence to SLE_{6}.

References

[1] M. Aizenman, A. Burchard, Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 (1999), 419–453.

[2] B. Bollobás, O. Riordan, A short proof of the Harris-Kesten theorem.

Preprint, arXiv:math.PR/0410359.

[3] F. Camia, C. Newman, The full scaling limit of two-dimensional critical percolation. Preprint, arXiv:math.PR/0504036.

[4] J. L. Cardy, Conformal invariance and surface critical behaviour, Nucl. Phys. B240 (1984), 514–532.

[5] G. Grimmett, Percolation, 2nd ed. Springer, 1999.

[6] H. Kesten, Scaling relations for 2D-percolation Comm. Math. Phys. 109 (1987), 109–156.

[7] H. Kesten, V. Sidoravicius, Y. Zhang, Almost all words are seen in critical site percolation on the triangular lattice. Electr. J. Prob. 3 (1998), paper no. 10.

[8] R. Langlands, P. Pouliot, Y. Saint-Aubin, Conformal invariance in twodimensional percolation. Bull. A.M.S. 30 (1994), 1–61.

[9] G. F. Lawler, O. Schramm, W. Werner, One-arm exponent for critical 2D percolation. Electr. J. Pobab. 7 (2002), paper no. 2.

[10] O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288.

[11] S. Smirnov, Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris 333 (2001), 239–244. Expanded version available at http://www.math.kth.se

    /~stas/papers/percol.ps

[12] S. Smirnov, W. Werner, Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (2001), 729–744.

Gábor Pete University of California, Berkeley email: gabor@stat.berkeley.edu

12 Conformal invariance of planar loop-erased random walks and uniform spanning trees Part II — UST

after G. Lawler, O. Schramm, and W. Werner [1] as summarized by Peter Ralph

Abstract The scaling limit of the Peano curve associated with the uniform spanning tree (UST) on a lattice approximation of a C^{1} simply connected domain, wired to a connected subarc of the boundary, is shown to be chordal SLE(8).

12.1 Introduction

The convergence of loop–erased random walk (LERW) to SLE(2) implies, via Wilson’s algorithm, existence of a conformal scaling limit for the uniform spanning tree on a lattice approximation of a simply connected, C^{1} domain. There is another connection between the UST and SLE. Using similar techniques to the proof that LERW converges to radial SLE(2), it can be shown that the curve that winds between the UST and its’ planar dual (called the UST Peano curve) converges to chordal SLE(8), which is therefore a random, space-filling curve. Although the proofs are written with the lattice approximation living on ℤ^{2}, the proofs can be adapted to more general grids.

The proof first shows convergence of the driving process for the discrete path to B(8t), and then uses this fact and harmonic function estimates to show tightness, and hence weak convergence, of the measures on curves. The proof of driving process convergence makes use of an “observable” quantity (a hitting probability) whose conditional expectation with respect to the first segment of the Peano path gives information about the trajectory of the Peano path from that point, information that can be translated into a relationship between movement of the driving process and capacity of the path from infinity.

12.2 Preliminaries

Weak convergence of curves are with respect to the metric ρ on unparameterized paths [0, 1] → ℂ, defined by ρ(β, γ) = inf sup_{t∈[0,1]}{|β̂(t) − γ̂(t))|}, where the infimum is over reparameterizations of the paths.

Let 𝕌 denote the open unit disc in ℂ.

12.3 UST and the Peano curve

A spanning tree on a finite graph G is a connected subgraph with no cycles that contains every vertex in G. The uniform spanning tree on a finite graph is the uniform (counting) measure on all such spanning trees. Wilson’s al-

gorithm [3] (or see wikipedia.org/wiki/Uniform spanning tree for a de-

scription) makes a close relationship between UST and LERW, which allows us to prove things about the UST.

Suppose we have a planar graph G that is obtained in a natural way by approximating a simply connected domain D ⊂ ℂ with a subgraph of some lattice. Say that two points a and b can be removed to partition the boundary into two connected pieces, α and β, and we condition the UST to contain the lattice edges in the boundary of G near α, but put no particular conditioning on those near β. The resultant random tree is called the UST with wired boundary near α and free boundary near β. Each edge in the lattice has associated a unique edge in the dual lattice, and so it is easy to see that the edges not used in a given spanning tree induce a “dual” spanning tree on (some subgraph of) the dual lattice. If we choose the appropriate approximation to D in the dual lattice, there is for each spanning tree wired near α and free near β, a dual spanning trees wired near β and free near α, and these two trees do not intersect.

Since the two trees do not intersect, we can draw a curve between them, a curve that starts near a and ends near b, and passes near every vertex in G. This is the Peano curve associated with a spanning tree. The measure on such curves induced by the UST on G is the UST Peano curve. We haven’t made precise where, exactly, the curve (or for that matter, the dual lattice) goes, but this isn’t difficult — for instance, say that our lattice is ℤ^{2}, the dual lattice is ℤ^{2} + 1/2, and the curve between them is on ℤ^{2}/2 + 1/4.

This is the curve that converges weakly to SLE(8) in the scaling limit. For each δ > 0,let G_{δ} be a (suitably defined) approximation to D in δℤ^{2}.

Theorem 1 (UST Peano path scaling limit). The UST Peano curve in G_{δ} with wired boundary near α and free boundary near β converges weakly as δ → 0 to the image of chordal SLE(8) under any conformal map from ℍ to D that takes 0 to a and ∞ to b. The weak convergence is with respect to the metric ρ.

When we refer to “the UST” below, we are referring to the UST with these boundary conditions.

12.4 Convergence of discrete harmonics

The proof will rest on the following estimate for how close a discrete harmonic function is to its’ continuous counterpart at a fixed point (this point will be zero), uniformly over large enough domains, and is indicative of the sort of harmonic estimates are involved. If instead of requiring the inradius of D at zero to be large enough, we instead fix the inradius and require the mesh to be fine enough, it says that the probability that SRW begun at zero and reflected off some subarc of the boundary exits through a subarc of the rest of the boundary is close to the corresponding continuous harmonic function, uniformly over choices of domain, subarc, and small enough mesh size.

Recall that a discrete harmonic function on a graph is equal at each vertex to the average over the neighbors of that vertex. Let H be a finite nonempty connected subgraph of ℤ^{2}, and let E_{∂} be it’s edge boundary. Let E_{∂} = E_{0} ∪E_{1} ∪E_{2} partition E_{∂}, with E_{0} ∪E_{1} ≠ ∅. Let ĥ = ĥ(H, E_{0}, E_{1}, E_{2}) : v ∈ V (H) → [0, 1] be the probability that SRW in ℤ^{2}\E_{2} begun at v will exit H through an edge in E_{1}. ĥ is also the unique function on H ∪ E_{∂} (if we give each edge in E_{∂} a new vertex, so no two share the same endpoint outside H) that is harmonic in H, has boundary conditions ĥ = 0 on E_{0}, ĥ = 1 on E_{1}, and ĥ(u) = ĥ(v) across edges (u, v) in E_{2}, which makes it plausible that the estimate is true. Recall that “Neumann boundary conditions” on some piece of the boundary means that the outward normal derivative is zero along that piece of the boundary. Let ψ_{D} be the conformal map that takes D to 𝕌 and fixes zero.

Theorem 2 (Dirichlet-Neumann approximation). For every ϵ > 0 there is an r_{0} = r_{0}(ϵ) such that the following holds. Let D ⊂ ℂ be a simply connected domain with rad_{0}(D) ≥ r_{0}. Let A_{0}, A_{1} ⊂ ∂𝕌 be two disjoint arcs, each of length at least ϵ, and set A_{2} = ∂𝕌 \ (A_{0} ∪ A_{1}), and also pull back this partition to the boundary of D using ψ_{D}. Let H be a lattice approximation of D in ℤ^{2}. Let h : 𝕌 → [0, 1] be the harmonic function in 𝕌 that has boundary value 0 on A_{0}, 1 on A_{1}, and Neumann boundary conditions on A_{2}. Let ĥ = ĥ(H, A_{0}, A_{1}, A_{2}) the discrete harmonic function as defined above. Then |ĥ(0) − h(0)| < ϵ.

Note that we are approaching the scaling limit by dilating the domain rather than shrinking the lattice spacing, which is clearly equivalent in this case.

12.5 Driving process convergence

Since the UST Peano path γ_{t} in a domain is a continuous path between two boundary points, there exists a driving process that through the chordal Loewner equation (and a conformal map D → ℍ) has γ_{t}, reparametrized by capacity from infinity, as it’s hull. The first step in showing weak convergence is to show convergence of the driving process to suitably scaled Brownian motion. The convergence follows fairly straightforwardly from the following estimate, using Skorokhod’s embedding, a step we omit.

For a domain D containing zero with boundary points a and b, let ϕ_{0} be the conformal map from D to ℍ taking a to 0, b to ∞, and with |ϕ(0)| = 1. Let γ be the UST Peano path in D, and let ϕ_{n} be the conformal map from D_{k} := D \ γ[0, n] to ℍ so that ϕ_{n} ∘ ϕ_{0}^{−1} has the hydrodynamic normalization. The quantities of interest are W_{n} = ϕ_{n}(γ(n)) and t_{n}, the capacity of ϕ_{0}(γ[0, n]) from infinity in ℍ.

Theorem 3 (Key estimate). For every sufficiently small δ, ϵ > 0 there is some r_{0} = r_{0}(δ, ϵ) such that the following holds. Let k ∈ ℕ and let m ≥ k be the first n ≥ k such that |W_{n} − W_{k}| ≥ δ or t_{n} − t_{k} ≥ δ^{2}. Then

𝔼[W_{m} − W_{k}|D_{k}] = O(δ^{3}), and
𝔼[(W_{m} − W_{k})^{2}|D_{k}] − 8𝔼[t_{m} − t_{k}|D_{k}] = O(δ^{3}),

if rad_{0}(D_{k}) ≥ r_{0} and the harmonic measure in D_{k} at zero of each piece of the boundary between b and γ(n) is at least ϵ.

The proof works by calculating the approximate change in the expected value of an “observable” related to the UST as we condition on a longer segment of the UST Peano curve. Since the change can be well–approximated by the change in W_{n} and t_{n}, the martingale property implies the equations of the theorem that relate the two. A sufficiently large subarc A of the boundary arc α and a suitable vertex well in the interior of D are fixed, and the observable is the event A that the path in the UST from this vertex to α hits α on A. The probability that a Brownian path reflected off the negative real axis first exits ℍ through [0, 1] is calculable, and so using this calculation, the map ϕ_{n}, and Theorem 2, we can the approximate the conditional probability of this event given the first n steps of the UST Peano curve — which, intuitively, should encode information about the trajectory of the Peano curve. The chordal Loewner equation implies that the first–order change in the image of a point, say v, at time 0 in ℍ under ϕ_{n} ∘ ϕ_{0}^{−1} is 2t_{n}/v, provided t_{n} is small enough, and v stays far enough from the boundary. This allows us to expand the martingale equation ℙ[A] = 𝔼[ℙ[A|D_{m}]] to first order in the image of the reference point, first order in t_{m}, and second order in W_{m}. A few convenient choices of reference point lead to the equations of the theorem.

12.6 Stronger convergence

To prove weak convergence with respect to ρ, tightness of the laws of the UST Peano curves on successively finer grids is needed, which is shown via a uniform continuity estimate (and Arzela-Ascoli). For a domain D and positive number R, let D^{R} be a lattice approximation of its’ dilation RD, let γ_{R} be the UST Peano path in D^{R}, and let γ̂_{R} be the image of γ_{R} under the conformal map from D^{R} to ℍ with the usual normalization.

Theorem 4 (Uniform continuity estimate). For every ϵ > 0 and t̄ > 0 there are some positive R_{0} = R_{0}(D, t̄, ϵ) and δ = δ(D, t̄, ϵ) such that for all R > R_{0},

ℙ[sup{|γ̂_{R}(t_{2}) − γ̂_{R}(t_{1})| : t_{1}, t_{2} ∈ [0, t̄], |t_{2} − t_{1}| ≤ δ} > ϵ] < ϵ.

The proof is involved, and makes use of estimates derived in [2]. It depends mostly on the use of Wilson’s algorithm to relate harmonic measure estimates to the behavior of the UST.

References

[1] Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees. arXiv:math.PR/0112234v3.

[2] Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees. arXiv:math.PR/9904022v2.

[3] David Wilson, Generating random spanning trees more quickly than the cover time. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp 296–303, New York, 1996. ACM.

Peter Ralph, UC Berkeley email: plr[at]stat.berkeley.edu

13 Conformal restrictions, highest weights and SLE

after R. Friedrich and W. Werner A summary written by Shannon Starr

Abstract

We review the Virasoro algebra and then (most of) [1].

13.1 Background : the Virasoro Algebra

Here we will review some basics of the theory of highest weight representations for the Virasoro algebra, following [2].

13.1.1 Basic Definitions

Definition 1. The Virasoro algebra is the Lie algebra Vir with generators {d_{n} , n ∈ ℤ; c}, such that [d_{n}, c] = 0 and

[d_{m}, d_{n}] = (m − n)d_{m+n} + δ_{m,−n} \frac{(m^{3} − m)}{12}c .  (1)

A good way to think about Vir is to first consider another Lie algebra, A, called the Witt algebra which Vir reduces to if one sets c = 0. This is the Lie algebra of complex vector fields on the circle. It has the basis

X_{n} = i exp(inθ) \frac{d}{dθ} = −z^{n+1} \frac{d}{dz} (n ∈ ℤ) .

It is an exercise to check [X_{m}, X_{n}] = (m − n)X_{m+n}. There is an important map ω : A → A which is an antilinear anti-involution determined by

ω(X_{n}) = X_{−n} , ω(λX) = \overline{λ}ω(X) .

It has the feature that if X is real then ω(X) = −X.

The real vector fields can be thought of as the infinitesimal generators for the group of orientation preserving diffeomorphisms of the circle. This group G acts on smooth functions by π(ϕ)f(z) = f(ϕ^{−1}z). Taking ϕ “close to the identity”

ϕ(z) = z(1 + ϵ(z)) = z + ∑_{n∈ℤ} ϵ_{n}z^{n+1} ,

leads to the first order approximation for the action

π(ϕ)f(z) = f(z − ∑_{n∈ℤ} ϵ_{n}z^{n+1}) = (1 + ∑_{n∈ℤ} ϵ_{n}X_{n})f(z) ,

which shows the relation to A. This example illustrates the fact that any complex representation for G leads naturally to a representation of A. If V is such a representation, and if V possesses a Hermitian form such that

⟨ϕ(u)|ϕ(v)⟩ = ⟨u|v⟩ for all ϕ ∈ G , u, v ∈ V ,

then it is an easy consequence that

⟨Xu|v⟩ = ⟨u|ω(X)v⟩ for all X ∈ A , u, v ∈ V .

We call contravariant a Hermitian form satifying this.

The relationship between Vir and A is that Vir is a central extension of A, upon identifying (X_{n} : n ∈ ℤ) with (d_{n} : n ∈ ℤ). More precisely it is a central extension by a 1-dimensional center, ℂc. The new element c is called the central charge. In our application, we will be primarily interested in A. But we will first describe the representations of Vir. The antilinear anti-involution is extended to Vir by setting: ω(d_{n}) = d_{−n}, ω(c) = c .

13.1.2 Highest weight representations

The element d_{0} ∈ Vir is called the energy operator. It generates a 1dimensional subalgebra, ℂd_{0}. If d_{0} can be diagonalized in a representation V, we say V is graded by d_{0}. From the commutation relations, if V_{h} is the eigenspace of d_{0} with eigenvalue h, then d_{n}V_{h} ⊂ V_{h−n} for n ∈ ℤ.

Definition 2. A highest weight representation of Vir is a representation in a vector space V which admits a nonzero vector v such that for given complex numbers c, h:

c(v) = c v , d_{0}(v) = h v ,

and V is the linear span of vectors of the form

d_{−i_{k}} ⋯ d_{−i_{1}}(v) (0 < i_{1} ≤ ⋯ ≤ i_{k}) .  (2)

For a highest weight representation V, the pair (c, h) is called the highest weight, and v is called the highest weight vector. Obviously V is graded, and all vectors of the form (2) with a fixed value n = i_{1} + i_{2} + ⋯ + i_{k} span the eigenspace V_{h+n}. These vectors are naturally indexed by partitions of n. They may not all be linearly independent. Consideration of the definition shows that

d_{n}(v) = 0 for n > 0 .  (3)

Definition 3. A highest weight representation such that all the vectors of the form (2) are linearly independent is a Verma representation.

Any Verma representation has the universal property that every other highest weight representation for the same weight is a quotient of it. In particular, the Verma representation for a highest weight (c, h) is unique when it exists. It always exists, by standard Lie algebra techniques. It is denoted M(c, h).

Let us collect some simple, but useful facts:

Proposition 4. (a) M(c, h) has a maximal proper subrepresentation J(c, h).

(b) Provided that c and h are real, M(c, h) carries a unique contravariant Hermitian form ⟨·|·⟩ such that ⟨v|v⟩ = 1.

(b) The eigenspaces of d_{0} are pairwise orthogonal with respect to ⟨·|·⟩.

(c) Ker ⟨·|·⟩ = J(c, h).

We will not prove these facts. Most are consequences of the fact that any subrepresentation of M(c, h) respects the grading ⨁_{n∈ℤ_{+}} M(c, h)_{h+n}.

The most important thing for us is to determine the Hermitian form ⟨·|·⟩ from (b). Let us demonstrate on some examples. We consider the orthogonal blocks M(c, h)_{h+n}. Then n = 0 corresponds to the trivial identity ⟨v|v⟩ = 1. The n = 1 block is spanned by d_{−1}v, and

⟨d_{−1}v|d_{−1}v⟩ = ⟨v|ω(d_{−1})d_{−1}v⟩ = ⟨v|d_{1}d_{−1}v⟩ .

But by the commutation relations and (3),

d_{1}d_{−1}v = [d_{1}, d_{−1}]v = 2d_{0}v = 2hv .

So ⟨d_{−1}v|d_{−1}v⟩ = 2h. As an exercise, the reader should check, for n = 2,

\begin{matrix}⟨d_{−2}(v)|d_{−2}(v)⟩ = 4h + \frac{c}{2} , ⟨d_{−1}^{2}(v)|d_{−1}^{2}(v)⟩ = 8h^{2} + 4h , \\ ⟨d_{−2}(v)|d_{−1}^{2}(v)⟩ = ⟨d_{−1}^{2}(v)|d_{−2}(v)⟩ = 6h . \end{matrix}  (4)

13.1.3 Degenerate representations and det_{n}(c, h).

Let V (c, h) := M(c, h)/J(c, h) be the unique irreducible highest weight representation for weight (c, h). It is called degenerate if J(c, h) is nontrivial. If n is the smallest n ∈ ℤ_{+} such that J(c, h) ∩ M(c, h)_{h+n} is nontrivial, then V (c, h) is called degenerate at level n. Recall J(c, h) = Ker⟨·|·⟩. We define det_{n}(c, h) to be the determinant of the matrix representation (Gram matrix) of ⟨·|·⟩ with respect to the basis vectors (2) in the subspace M(c, h)_{h+n}. Clearly, J(c, h) ∩ M(c, h)_{h+n} is nontrivial if and only if det_{n}(c, h) = 0.

As an example, det_{1}(c, h) = det(⟨d_{−1}v|d_{−1}v⟩) = 2h. Therefore, V (c, h) is degenerate at level 1 if and only if h = 0. But this is a trivial representation: d_{n}v = 0 for all n. As a second example, using (4),

det_{2}(c, h) = |\begin{matrix}⟨d_{−2}v|d_{−2}v⟩ & ⟨d_{−2}v|d_{−1}^{2}v⟩ \\ ⟨d_{−1}^{2}v|d_{−2}v⟩ & ⟨d_{−1}^{2}v|d_{−1}^{2}v⟩ \end{matrix}| = |\begin{matrix}4h + \frac{c}{2} & 8h^{2} + 4h \\ 8h^{2} + 4h & 6h \end{matrix}| .

We leave it as an exercise to carry out the arithmetic, and in particular, check that V (c, h) is degenerate at level 2 if and only if h is one of the two values

h_{±}(c) = \frac{15 − 3c ± 3\sqrt{(c − 1)(c − 25)}}{48} (but not 0) .  (5)

13.2 Chordal Restriction Measures and Highest Weight Representations

13.2.1 Prologue

Let g_{t} : ℍ \ γ[0, t] → ℍ be a chordal SLE_{κ} process, such that g_{t}(γ_{t}) = \sqrt{κ}b_{t}, and let f_{t}(z) = g_{t}(z) − \sqrt{κ}b_{t}. (So dg_{t}(z) = dt/f_{t}(z).) Given a function F (x_{1}, …, x_{n}) on ℝ^{n}, let F_{t} := F (f_{t}(x_{1}), …, f_{t}(x_{n})). By Itô’s formula,

dF_{t} = [\sqrt{κ}L_{−1} + dt(\frac{κ}{2}L_{−1}^{2} − 2L_{−2})]F (f_{t}(x_{1}), …, f_{t}(x_{n})) .

where we have defined L_{N} := − ∑_{k=1}^{n} x_{k}^{1+N}∂_{k} for N = −1, −2. But, if we extend the definition to all N ∈ ℤ, then we have a representation of Vir with c = 0, according to (1). If, additionally, F_{t} is a local martingale, then

[\frac{κ}{2}L_{−1}^{2} − 2L_{−2}]F = 0 .

This is reminiscent of the condition to have a degenerate representation at level 2, if only F were a highest weight vector. The goal is to combine this

Figure 2: The event E = E_{ε}(x_{1}) ∩ E_{ε}(x_{2}) ∩ E_{ε}(x_{3}). (Copied from [1].)

type of observation with a rigorous construction of highest weight representations associated to 1-sided chordal restriction measures.

13.2.2 Boundary Correlation Functions

Let ℙ describe a 1-sided chordal restriction measure, with exponent h. (See the summary for topic 10, by Jesse Goodman.) Let β be the curve forming the right boundary of the set whose law is ℙ. For each x > 0 and ε > 0 one defines the event E_{ε}(x) = {β : β ∩ (x, x + iε] ≠ ∅}. One then defines the boundary correlation functions: B_{0} = 1 and for n > 0

B_{n}(x_{1}, …, x_{n}) = lim_{ε_{1},...,ε_{n}→0} ε_{1}^{−2} ⋯ ε_{n}^{−2}ℙ(E_{ε_{1}}(x_{1}) ∩ ⋯ ∩ E_{ε_{n}}(x_{n})) .

There is a remarkable identity for these functions:

Proposition 5 (Ward identity). For all n ∈ ℤ_{+} and x, x_{1}, …, x_{n} ∈ (0, ∞),

B_{n+1}(x, x_{1}, …, x_{n}) = (\frac{h}{x^{2}} +∑_{k=1}^{n} [\frac{2}{(x_{k} − x)^{2}} −(\frac{1}{x}+\frac{1}{x_{k} − x})∂_{x_{k}}])B_{n}(x_{1}, …, x_{n}) .

(6)

Before sketching why this is true in general, consider the special case of n = 0. The content of (6) is then that B_{1}(x) = h/x^{2}. The Riemann map from ℍ \ (x, x + iε\sqrt{2}] to ℍ is (with the correct normalization)

φ_{x,ε}(z) = \sqrt{(z − x)^{2} + 2ε^{2}} − \sqrt{(+i0 − x)^{2} + 2ε^{2}} ,

where √ : ℂ \ [0, ∞) → ℍ. Then, by the chordal restriction property, B_{1}(x) = lim_{ε→0} ε^{−2}(1 − [φ′_{x,ε}(0)]^{h}). One can check this is h/x^{2}.

Now, in order to see why (6) is true in general, consider the event (which has no x dependence) E_{ε} := E_{ε}(x_{1}) ∩ ⋯ ∩ E_{ε}(x_{n}). An obvious partition identity is

ℙ(E_{ε}) = ℙ(E_{ε} ∩ E_{δ}(x)) + [1 − ℙ(E_{δ}(x))] ℙ(E_{ε} | E_{δ}(x)^{c}) .

The left hand side is independent of x and δ. The trick is that there is some apparent dependence on x and δ on the right hand side, which must necessarily cancel. By carefully keeping track of all such terms to order δ^{2}, one obtains (6).

13.2.3 Highest Weight Representations of the Witt Algebra

One can treat (6) as a Laurent series in x. Define the operators

L_{N} = − ∑_{x_{k}}[2(N + 1)x_{k}^{N} + x_{k}^{N+1}∂_{k}] .

One should always specify which variables these operators are acting on (note the index of summation), but we rarely will. Then one can write

B_{n+1}^{(h)}(x, x_{1}, …, x_{n}) = [\frac{h}{x^{2}} + ∑_{N≥1} x^{N−2}L_{−N}]B_{n}^{(h)}(x_{1}, …, x_{n}) . (7)

Like the L_{N} operators, the L_{N} satisfy (1) for c = 0. However, we have still not yet obtained the desired highest weight representation.

For this purpose, consider a vector space of sequences of functions w = (w_{0}, w_{1}, w_{2}, …), where w_{n} = w_{n}(x_{1}, …, x_{n}) for each n ∈ ℤ_{+}. The primary example is B = (B_{0}, B_{1}, B_{2}, …). In analogy to (7), we define new vectors ℓ_{N}(w) for all N ∈ ℤ so that the following identity is true

w_{n+1}(x, x_{1}, …, x_{n}) = ∑_{N∈ℤ}x^{N−2}(ℓ_{−N}w)_{n}(x_{1}, …, x_{n}) .

For example, by (7): ℓ_{N}B = 0 for N > 0; ℓ_{0}B = hB; and ℓ_{N}B satisfies (ℓ_{N}B)_{n} = L_{N}B_{n} for N < 0.

Lemma 6. For all k ∈ ℤ_{+} and all N_{1}, …, N_{k} < 0, ℓ_{N_{1}} ⋯ ℓ_{N_{k}}B is the vector

(ℓ_{N_{1}} ⋯ ℓ_{N_{k}}B)_{n} = L_{N_{1}} ⋯ L_{N_{k}}B_{n} .

This is proved by a straightforward calculation using the definitions.

The purpose behind the lemma is to define a vector space V as the span of all such vectors (all N’s negative). Since ℓ_{N}B = 0 for all N > 0, this is a likely candidate for a highest weight representation of Vir with highest weight (0, h) and highest weight vector B. The operators representing the d_{n}’s are the ℓ_{N}’s. The following proposition asserts that this works.

Proposition 7. For any v ∈ V and every N ∈ ℤ, ℓ_{N}v ∈ V. Moreover, [ℓ_{N}, ℓ_{M}]v = (N − M)ℓ_{M+N}v for every M ∈ ℤ.

We will be content to verify that, for N ≥ 0 and M < 0, ℓ_{N}ℓ_{M}B ∈ V and to check that [ℓ_{N}, ℓ_{M}]B = (N − M)ℓ_{N+M}B.

By Lemma 6 we have (ℓ_{M}B)_{n} = L_{M}B_{n}. In order to calculate ℓ_{N}ℓ_{M}B by the definition, we really need to know (ℓ_{M}B)_{n+1}(x, x_{1}, …, x_{n}). Of course this is L_{M}B_{n+1}(x, x_{1}, …, x_{n}). We start with the formula

B_{n+1}(x, x_{1}, …, x_{n}) = ∑_{R≥0} x^{R−2}(ℓ_{−R}B)_{n}(x_{1}, …, x_{n}) .

Therefore, now indicating the variables L_{M} acts on in the superscript,

L_{M}B_{n+1}(x, x_{1}, …, x_{n}) = ∑_{R≥0} [(L_{M}^{(x)}x^{R−2})+x^{R−2}L_{M}^{(x_{1},...,x_{n})}](ℓ_{R}B)_{n}(x_{1}, …, x_{n}) .

(8)

For N > 0 the second term has no x^{−2−N} coefficient. An obvious calculation gives L_{M}^{(x)}x^{R−2} = −(2M − R)x^{−2−(R−M)}. So the coefficient of x^{−2−N} in the first term is: 0 if N > −M; or −(2M − R)(ℓ_{R}B)_{n}(x_{1}, …, x_{n}) if N = R − M for some R ≤ 0. In the latter case, we obtain (N −M)(ℓ_{N+M}B)_{n}(x_{1}, …, x_{n}). So ℓ_{N}ℓ_{M}B = (N − M)ℓ_{N+M}B. This is a vector in V, and since ℓ_{N}B = 0, the commutation relation is verified. The N = 0 calculation is similar, except one has to keep track of both terms in (8). This is also what is necessary for the commutation relation to be verified for N = 0.

The verification of the rest of the proposition uses essentially similar ideas.

13.2.4 The Markovian Property and Degeneracy

Now that we have a highest weight representation for Vir (actually A), let us return to the ideas in Section 13.2.1. Suppose that β corresponds to a SLE_{κ} curve for some κ. By the Markovian property of SLE_{κ}, the conditional probability that β hits the straight slits (x_{k}, x_{k} + iε\sqrt{2}], given β up to time t, is equal to the probability that β hits the curved slits f_{t}((x_{k}, x_{k} + iε\sqrt{2}]).

For ε → 0 these are well approximated by straight slits (this is also key to the proof of Proposition 5). So, dividing by ε^{2n} the asymptotic probability density is

f′_{t}(x_{1})^{−2} ⋯ f′_{t}(x_{n})^{−2}B_{n}^{(h)}(f_{t}(x_{1}), …, f_{t}(x_{n})) .  (9)

Because these are conditional probabilities, for t > 0, they form a local martingale. From Section 13.2.1 we already calculated

dB_{n,t} = \sqrt{κ}L_{−1}B_{n,t} db_{t} + (\frac{κ}{2}L_{−1}^{2} − 2L_{−2})B_{n,t} dt ,

where B_{n,t} = B_{n}(f_{t}(x_{1}), …, f_{t}(x_{n})). Moreover, since dg_{t}(z) = dt/f_{t}(z), d[f′_{t}(z)] = −f′_{t}(z)dt/[f_{t}(z)]^{2}. Therefore the differential of (9) is

∏_{k=1}^{n} f′_{t}(x_{k})^{−2}[db_{t} \sqrt{κ}L_{−1} + dt (\frac{κ}{2}L_{−1}^{2} − 2L_{−2} + ∑_{k=1}^{n} \frac{4}{f_{t}(x_{k})^{2}} )]B_{n,t} .

Since this is a local martingale, the drift must vanish. This condition can be precisely rewritten as

[\frac{κ}{2}L_{−1}^{2} − 2L_{−2}]B_{n}(f_{t}(x_{1}), …, f_{t}(x_{n})) = 0 ,  (10)

almost surely (by relating L_{−1} and L_{−2} to L_{−1} and L_{−2}). In other words, the highest weight representation is degenerate at level 2. Since c = 0 this could happen only for the values

h = h_{±}(c = 0) = \frac{15 ± 3\sqrt{(−1)(−25)}}{48} .

But h_{−} = 0, and this is disallowed for chordal restriction measures. So, for consistency, we must have h = 5/8. Going a bit further, we can determine that the kernel of ⟨·|·⟩ at level 2 is spanned by \frac{3}{2}d_{−2}v − d_{−1}^{2}v. So, for consistency with (10), we must have κ/4 = 2/3: in other words, κ = 8/3.

References

[1] R. Friedrich and W. Werner, Conformal restrictions, highest weight representations, and SLE, Commun. Math. Phys., 243 (2003), 105-122.

[2] V. G. Vac and A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, 1987.

Shannon Starr, University of California, Los Angeles email: sstarr@math.ucla.edu