EXISTENCE OF MULTI-POINT BOUNDARY GREEN'S FUNCTION FOR CHORDAL SCHRAMM-LOEWNER EVOLUTION (SLE) By Rami Fakhry A DISSERTATION Submitted to Michigan State University in partial fulllment of the requirements for the degree of Mathematics  Doctor of Philosophy 2021 ABSTRACT EXISTENCE OF MULTI-POINT BOUNDARY GREEN'S FUNCTION FOR CHORDAL SCHRAMM-LOEWNER EVOLUTION (SLE) By Rami Fakhry Schramm-Loewner evolution (SLE for short) is a one-parameter (κ ∈ (0, 8)) family of random fractal curves which grow in plane domains. For an SLEκ curve γ growing in a domain D, and a single point z0 ∈ D or ∈ ∂D, the Green's function for γ at z0 is the limit G(z) := lim+ r−α P[dist(z0 , γ) ≤ r] r→0 for some suitable exponent α > 0, provided that the limit exists and is not trivial. The Green's function for SLE plays an central role in determining the Hausdor dimension of SLE, and proving the existence of Minkowski content of SLE. The notion of (one-point) Green's function easily extends to multi-point Green's function. Given n distinct points z1 , . . . , zn ∈ D or ∈ ∂D, the n-point Green's function for the SLEκ curve γ at (z1 , . . . , zn ) is the limit Yn G(z1 , . . . , zn ) := lim rj−α P[dist(zj , γ) ≤ rj , 1 ≤ j ≤ n]. r1 ,...,rn →0+ j=1 In the thesis, we prove that the n-point Green's function exists if γ is a chordal SLE, 8 κ ∈ (0, 8), α = κ − 1, z1 , . . . , zn ∈ ∂D, and ∂D is smooth near each zj . In addition, we prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function. In memory of Rami Fakhry and dedicated to his family. iii TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . v CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . .. . . . . . . . . . . 1 CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . .. . . . . . . . . . . 6 2.1 Symbols and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 H-Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Chordal Loewner Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Chordal SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Extremal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Two-sided Chordal SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 CHAPTER 3 MAIN ESTIMATES . . . . . . . . . . . . . . . . .. . . . . . . . . 15 CHAPTER 4 PROOF OF THE MAIN THEOREM . . . . . . . . . . . . . . . . . 35 4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 45 iv LIST OF FIGURES Figure 3.1: A gure for the proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . 20 Figure 3.2: The rst gure for the proof of Lemma 3.5 . . . . . . . . . . . . . . . . . 22 Figure 3.3: The second gure for the proof of Lemma 3.5 . . . . . . . . . . . . . . . 24 Figure 3.4: A gure for the proof of Lemma 3.7 . . . . . . . . . . . . . . . . . . . . . 26 Figure 3.5: A gure for the proof of Lemma 3.10 . . . . . . . . . . . . . . . . . . . . 34 v CHAPTER 1 INTRODUCTION The Schramm-Loewner evolution (SLE for short) is a one-parameter (κ ∈ (0, ∞)) family of random fractal curves which grow in plane domains. It was dened in the seminal work of Schramm [20] in 1999. Because of its close relation with two-dimensional lattice models, Gaussian free eld, and Liouville quantum gravity, SLE has attracted a lot of attention for over two decades. The geometric properties of an SLEκ curve depends on the parameter κ. When κ ≥ 8, an SLEκ curve visits every point in the domain (cf. [19]); when κ ∈ (0, 8), an SLEκ curve has κ Hausdor dimension d0 = 1 + 8 (cf. [4]). There are several types of SLE. In this thesis we focus on chordal SLE, which grows in a simply connected plane domain from one boundary point to another boundary point. Suppose γ is a chordal SLEκ curve, κ ∈ (0, 8), in a domain D, and z0 ∈ D. The Green's function for γ at z0 is the limit G(z0 ) := lim+ r−α P[dist(z0 , γ) ≤ r] (1.1) r→0 for some suitable exponent α > 0 depending on κ such that the limit exists and is not trivial, i.e., lies in (0, ∞). This notion easily extends to n-point Green's function: Yn G(z1 , . . . , zn ) := lim rj−α P[dist(zj , γ) ≤ rj , 1 ≤ j ≤ n], (1.2) r1 ,...,rn →0+ j=1 where z1 , . . . , zn are distinct points in D, provided that the limit exists and is not trivial. The term Green's function is used for the following reasons. Recall the Laplacian Green's function GD (z, w), z ̸= w ∈ D, for a planar domain D, which is characterized by the following properties: for any w ∈ D, ˆ GD (·, w) is positive and harmonic on D \ {w}. ˆ As z → ∂D, GD (z, w) → 0. 1 ˆ As z → w, GD (z, w) = − 2π 1 ln |z − w| + O(1). One important fact is − ln r w G(z, w) = lim+ · P [dist(z, B[0, τD ]) ≤ r], (1.3) r→0 2π where B is a planar Brownian motion started from w, and τD is the exit time of D. Notice the similarity between (1.1) and (1.3). The main dierence between them is the normalization − ln r factor: one is r−α and the other is 2π . Another important fact is: for any measurable set U ⊂ D, Z w E [|{t ∈ [0, τD ) : Bt ∈ U }|] = G(z, w)dA(z). (1.4) U Here |·| stands for the Lebesgue measure on R, and A is the area. It turns out that the correct exponent α is the co-dimension of the SLE curve, i.e., α = 2 − d0 = 2 − (1 + κ8 ) = 1 − κ8 . The existence of a variation of Green's function for chordal SLEκ , κ ∈ (0, 8), was given in [5], where the conformal radius was used instead of Euclidean distance. The existence of 2-point Green's function was proved in [12] (again for conformal radius instead of Euclidean distance) following a method initiated by Beara [4]. In [8] the authors showed that Green's function as dened in (1.2) (using Euclidean distance) exists for n = 1, 2, and then used those Green's functions to prove that ˆ an SLEκ curve γ can be parametrized by its d0 -dimensional Minkowski content, i.e., for any t1 < t2 , the (1 + κ8 )-dimensional Minkowski content of γ[t1 , t2 ] is t2 − t1 ; and ˆ under such parametrization, for any measurable set U ⊂ D, Z E[|{t : γ(t) ∈ U }|] = G(z)dA(z). (1.5) U The Minkowski content parametrization agrees with the natural parametrization introduced earlier (cf. [11, 13]) The similarity between (1.4) and (1.5) further justies the terminology Green's function. 2 In a series of papers ([17, 16]) the authors showed that the Green's function of chordal SLE exists for any n ∈ N. In addition, they found convergence rate and modulus of continuity of the Green's functions, and provided up-to-constant sharp bounds for them. If the reference point(s) is (are) on the boundary of the domain instead of the interior, we may use (1.1) and (1.2) to dene the one-point and n-point boundary Green's function. Again, if κ ≥ 8, the boundary Green's function makes no sense for SLEκ since it visits every point on the boundary; if κ ∈ (0, 8), the intersection of the SLEκ curve with the boundary 8 has Hausdor dimension d1 = 2 − κ ([3]), and so the reasonable choice of the exponent α is 8 α = 1 − d1 = κ − 1. Greg Lawler proved (cf. [6]) the existence of the 1- and 2-point (on the same side) bound- ary Green's function for chordal SLE, and used them to prove that the d1 -dimensional Minkowski content of the intersection of SLEκ with the domain boundary exists. He also obtained the exact formulas of these Green's function up to some multiplicative constant. We will use the exact formula of the one-point Green's function: G(z) = b c|z|−α , z ̸= 0, (1.6) where c>0 b is some (unknown) constant depending only on κ. We will also use the con- vergence rate of the one-point Green's function ([6, Theorem 1]): there is some constant C, β1 > 0 depending only on κ such that for any z ∈ R \ {0} and r ∈ (0, |z|), |P[dist(z, γ) ≤ ε] − G(z)εα | ≤ C(ε/|z|)α+β1 . (1.7) To the best of our knowledge, the existence of n-point boundary Green's function for n > 2 and the 2-point boundary Green's function when the two reference points lie on dierent sides of 0 has not been proved so far. The main goal of this thesis is to prove this existence for all n∈N without assuming that the reference points all lie on the same side of 0. In addition we prove that the Green's functions are continuous. We do not have exact formulas of these functions, but nd some sharp bounds for them in terms of simple functions. 3 We will mainly follow the approach in [16], and apply the results from there as well as from [6] and [17]. Below is our main result. Theorem 1.1. Let κ ∈ (0, 8) and α = κ8 −1. Let γ be an SLEκ curve in H := {z ∈ C : Im z > 0} from 0 to ∞. Let n ∈ N and Σn = {(z1 , . . . , zn ) ∈ (R \ {0})n : zj ̸= zk wheneverj ̸= k}. Then for any z = (z1 , . . . , zn ) ∈ Σn , the limit G(z) in (1.2) exists and lies in (0, ∞). Moreover, the convergence in (1.2) is uniform on each compact subset of Σn , the function G is continuous on Σn , and there is an explicit function F on Σn (dened in (3.4)) with a simple form such that G(z) ≍ F (z), where the implicit constants depend only on κ and n. Our result will shed light on the study of multiple SLE. For example, if we condition the chordal SLEκ in Theorem 1.1 to pass through small discs centered at z1 < z2 < · · · < zn ∈ (0, ∞), and suitably take limits while sending the radii of the discs to zero, then we should get an (n + 1)-SLEκ conguration in H with link pattern (0 ↔ z1 ; z1 ↔ z2 ; z2 ↔ z3 ; . . . ; zn−1 ↔ zn ; zn ↔ ∞), which is a collection of (n + 1) random curves (γ0 , . . . , γn ) in H such that γj connects zj with zj+1 , where z0 := 0 and zn+1 := ∞, and when any n curves among the (n + 1) curves are given, the last curve is a chordal SLEκ curve in a connected component of the complement of the given n curves in H. The n-point boundary Green's function is then closely related to the partition function associated to such multiple SLE. Here are a few topics that we could study in the near future. We may consider mixed multi-point Green's functions for chordal SLE, where some reference points lie in the interior of the domain, and some lie on the boundary. We expect that the Green's functions still 8 exist, if 1 − κ8 is used as the exponent for interior points and κ − 1 is used as the exponent for boundary points. We may also work on other types of SLE such as radial SLE, which grows from a boundary point to an interior point. The multi-point (interior) Green's function for radial SLE was proved to exist in [14]. The next natural objects to study are the boundary and mixed multi-point Green's function for radial SLE. The rest of the thesis is organized in a straightforward fashion. In Chapter 2, we recall symbols, notation and some basic results that are relevant to the thesis. Chapter 3 contains 4 the most technical part of the thesis, where we derive a number of important estimates. We nish the proof of the main theorem in Chapter 4. 5 CHAPTER 2 PRELIMINARIES 2.1 Symbols and Notation Let H = {z ∈ C : Im z > 0} be the open upper half plane. Given z0 ∈ C and S ⊂ C, we use radz0 (S) to denote sup{|z − z0 | : z ∈ S ∪ {z0 }}. We write Nn for {k ∈ N : k ≤ n}, where N = {1, 2, 3, . . . } is the set of all positive integers. For a, b ∈ R, we write a∧b and a∨b respectively for min{a, b} and max{a, b}. κ 8 We x κ ∈ (0, 8) and set d=1+ 8 and α= κ − 1. Throughout, a constant (such as α) depends only on κ and a variable n∈N (number of points), unless otherwise specied. We use X≲Y or Y ≳X if there is a constant C>0 such that X ≤ CY . We write X≍Y if X≲Y and Y ≲ X. When a (deterministic or random) curve γ(t), t ≥ 0, is xed in the context, we let τS = inf({t ≥ 0 : γ(t) ∈ S} ∪ {∞}). We write τrz0 for τ{z:|z−z0 ≤r} , and Tz0 for τ0z0 = τ{z0 } . So another way to say that dist(z0 , γ) ≤ r is τrz0 < ∞. We also write τR∞ for τ{z:|z|≥R} . A crosscut in a domain D is an open simple curve in D, whose two ends approach to two boundary points of D. When D is a simply connected domain, any crosscut ρ of D divides D into two connected components. 2.2 H-Hulls A relatively closed bounded subset K of H is called an H-hull if H\K is simply connected. The complement domain H\K is then called an H-domain. Given an H-hull K , we use gK to denote the unique conformal map from H\K onto H that satises gK (z) = z + O(|z|−1 ) −1 as z → ∞. Let f K = gK . The half-plane capacity of K is hcap(K) := limz→∞ z(gK (z) − z). If K = ∅, then gK = fK = id, and hcap(K) = 0. Now suppose K ̸= ∅. Let aK = min(K ∩ R) and bK = max(K ∩ R). Let K doub = K ∪ [aK , bK ] ∪ {z : z ∈ K}. By Schwarz reection 6 principle, gK extends to a conformal map from C \ K doub onto C \ [cK , dK ] for some cK < dK ∈ R, and satises gK (z) = gK (z). In this thesis, we write SK for [cK , dK ]. In the case K = ∅, we understand SK and [aK , bK ] as the empty set. Given two H-hulls K1 ⊂ K2 , we get another H-hull K2 /K1 dened by K2 /K1 = gK1 (K2 \ K1 ). Example 2.1. For x0 ∈ R and r > 0, the set K := {z ∈ H : |z − x0 | ≤ r} is an H-hull, r2 aK = x0 − r, bK = x0 + r, gK (z) = z + z−x0 , hcap(K) = r2 , and SK = [x0 − 2r, x0 + 2r]. Proposition 2.2. For any H-hull K , [aK , bK ] ⊂ SK . If K1 ⊂ K2 are two H-hulls, then SK1 ⊂ SK2 and SK2 /K1 ⊂ SK2 . Proof. This is [21, Lemmas 5.2 and 5.3]. Proposition 2.3. If a nonempty H-hull K satises that radx0 (K) ≤ r for some x0 ∈ R and r > 0, then hcap(K) ≤ r2 , SK ⊂ [x0 − 2r, x0 + 2r], and |gK (z) − z| ≤ 3r, z ∈ C \ K doub . (2.1) Moreover, for any z ∈ C with |z − x0 | ≥ 5r, we have  r 2 |gK (z) − z| ≤ 2|z − x0 | ; (2.2) |z − x0 | ′  r 2 |gK (z) − 1| ≤ 5 . (2.3) |z − x0 | Proof. This is [16, Lemmas 2.5 and 2.6]. Proposition 2.4. Let H be a nonempty H-hull, and H(H) denote the space of H-hulls, which are subsets of H . Then H(H) is compact in the sense that any sequence (Kn ) in H(H) contains a convergent subsequence (Knk ) whose limit K is contained in H(H). Here the convergence means that gKnk converges to gK locally uniformly in C \ H doub . Proof. This is [21, Lemma 5.4]. 7 2.3 Chordal Loewner Processes Let U (t), 0 ≤ t < T , be a real valued continuous function, where T ∈ (0, ∞]. The chordal Loewner equation driven by U is the equation 2 ∂t gt (z) = , g0 (z) = z. (2.4) gt (z) − Ut For every z ∈ C, let τz∗ denote the rst time that the solution g· (z) blows up; when such time does not exist, τz∗ is set to be ∞. Let Kt = {z ∈ H : τz∗ ≤ t}. We call gt and Kt , 0 ≤ t < T , the chordal Loewner maps and hulls, respectively, driven by U. It turns out that, for each t ∈ [0, T ), Kt is an H-hull, hcap(Kt ) = 2t, and gt = gKt . Proposition 2.5. For any 0 ≤ t < T , \ {Ut } = Kt+ε /Kt . ε∈(0,T −t) Proof. This a restatement of [9, Theorem 2.6]. Corollary 2.6. If for some H-hull H and t0 ∈ (0, T ), Kt0 ⊂ H , then Ut ∈ SH for 0 ≤ t < t0 . Proof. By Proposition 2.5, for every t ∈ [0, t0 ), Ut ∈ [aKt0 /Kt , bKt0 /Kt ], which implies by Proposition 2.2 that Ut ∈ SKt0 /Kt ⊂ SKt0 ⊂ SH . By the continuity of U, we also have Ut0 ∈ SH . We call the maps Zt = gt − Ut the centered Loewner maps driven by U. Proposition 2.7. Let b > a ∈ [0, T ). Suppose that radx0 (Kb /Ka ) ≤ r for some x0 ∈ R and r > 0. Then |Za (z) − Zb (z)| ≤ 7r for any z ∈ H \ Kb . Proof. Let Ua;t = Ua+t , ga;t = ga+t ◦ ga−1 , and Ka;t = Ka+t /Ka , 0 ≤ t < T − a. It is straightforward to check that ga;· and Ka;· are respectively the chordal Loewner maps and 8 hulls driven by Ua;· . By Corollary 2.6, Ua , Ub ∈ SKa;b−a . By the assumption, radx0 (Ka;b−a ) ≤ r. By Proposition 2.3, SKa;b−a ⊂ [x0 − 2r, x0 + 2r]. Thus, |Ua − Ub | ≤ 4r. By Proposition 2.3, |ga;b−a (z) − z| ≤ 3r for any z ∈ H \ Ka;b−a . So for any z ∈ H \ Kb , |ga (z) − gb (z)| ≤ 3r. Since Zt = gt − Ut , and |Ua − Ub | ≤ 4r, we get the conclusion. If there exists a function γ(t), 0 ≤ t < T , in H, such that for any t, H \ Kt is the unbounded connected component of H \ γ[0, t], we say that such γ is the chordal Loewner curve driven by U. Such γ may not exist in general, but when it exists, it is determined by U, and for each t ∈ [0, T ), gt−1 and Zt−1 extend continuously from H to H and satisfy gt−1 (Ut ) = Zt−1 (0) = γ(t). 2.4 Chordal SLE √ Let κ > 0. Let Bt be a standard Brownian motion. If the driving function is Ut = κBt , 0 ≤ t < ∞, then the chordal Loewner curve driven by U exists, starts from 0 and ends at ∞ (cf. [19]). Such curve is called a chordal SLEκ trace or curve in H from 0 to ∞. Its geometric property depends on κ: if κ ≤ 4, it is simple; if 4 < κ < 8, it is not simple and not space-lling; if κ ≥ 8, it is space-lling (cf. [19]). The Hausdor dimension of an SLEκ curve is min{1 + κ8 , 2} (cf. [19, 4]). The denition of chordal SLE extends to general simply connected domains via confor- mal maps. Let D be a simply connected domain with two distinct boundary points (more precisely, prime ends) a, b. Let f be a conformal map from H onto D, which sends 0 and ∞ respectively to a and b. Let γ be a chordal SLEκ curve in H from 0 to ∞. Then f ◦γ is called a chordal SLEκ curve in D from a to b. A remarkable property of SLE is the Domain Markov Property (DMP). Suppose γ is a chordal SLEκ curve in H from 0 to ∞, which generates the H-hulls Kt , 0 ≤ t < ∞, and a ltration F = (Ft )t≥0 . Let τ be a nite F -stopping time. Conditionally on Fτ , γ(τ + ·) has the same law of a chordal SLEκ curve in H \ Kτ from γ(τ ) to ∞. Equivalently, there is a chordal SLEκ curve γ e in H from 0 to ∞ independent of Fτ such that γ(τ + t) = Zτ−1 (e γ (t)), 9 t ≥ 0. Here Zτ is the centered Loewner map at the time τ that corresponds to γ, and its inverse Zτ−1 has been extended continuously to H. We will also use the left-right symmetry and rescaling property of chordal SLE. Suppose γ is a chordal SLEκ curve in H from 0 to ∞. The left-right symmetry states that, if f (z) = −z is the reection about iR, then f ◦ γ has the same law as γ . This follows easily from that √ √ (− κBt ) has the same law as ( κBt ). The rescaling property states that, for any c > 0, √ (cγ(t)) has the same law as (γ( ct)). This follows easily from the rescaling property of the Brownian motion. 2.5 Extremal Length We will need some lemmas on extremal length, which is a nonnegative quantity λ(Γ) asso- ciated with a family Γ of rectiable curves ([1, Denition 4-1]). One remarkable property of extremal length is its conformal invariance ([1, Section 4-1]), i.e., if every γ∈Γ is contained in a domain Ω, and f is a conformal map dened on Ω, then λ(f (Γ)) = λ(Γ). We use dΩ (X, Y ) to denote the extremal distance between X and Y in Ω, i.e., the extremal length of the family of curves in Ω that connect X with Y. It is known that in the special case when Ω is a semi-annulus {z ∈ H : R1 < |z − x| < R2 }, where x ∈ R and R1 > R1 > 0, and X and Y are the two boundary arcs {z ∈ H : |z − x| = Rj }, j = 1, 2, then dΩ (X, Y ) = log(R2 /R1 )/π ([1, Section 4-2]). We will use the comparison principle ([1, Theorem 4-1]): if every γ∈Γ contains a γ ′ ∈ Γ′ , then λ(Γ) ≥ λ(Γ′ ). Thus, if every curve in Ω connecting X with Y a semi-annulus with radii R1 , R2 , then dΩ (X, Y ) ≥ log(R2 /R1 )/π . We will also use the com- position law ([1, Theorem 4-2]): if for j = 1, 2, every γj in a family Γj is contained in Ωj , where Ω1 and Ω2 are disjoint open sets, and if every γ in another family Γ contains a γ1 ∈ Γ1 and a γ2 ∈ Γ2 , then λ(Γ) ≥ λ(Γ1 ) + λ(Γ2 ). The following propositions are applications of Teichmüller Theorem. Proposition 2.8. Let S1 and S2 be a disjoint pair of connected closed subsets of H that 10 intersect R such that S1 is bounded and S2 is unbounded. Let zj ∈ Sj ∩ R, j = 1, 2. Then radz1 (S1 ) 1∧ ≤ 32e−πdH (S1 ,S2 ) . |z1 − z2 | Proof. For j = 1, 2, let Sjdoub be the union of Sj and its reection about R. By reection principle ([1, Exercise 4-1]), dH (S1 , S2 ) = 2dC (S1doub , S2doub ). Let r = radz2 (S1 ), L = |z1 − z2 | and R = L/r. From Teichmüller Theorem ([1, Theorem 4-7]), dC (S1doub , S2doub ) ≤ dC ([−r, 0], [L, ∞)) = dC ([−1, 0], [R, ∞)) = Λ(R). From [1, Formula (4-21)], we have doub ,S doub ) 1 e−πdH (S1 ,S2 ) = e−2πdC (S1 2 ≥ e−2πΛ(R) ≥ . 16(R + 1) 1 2 Since 1∧ R ≤ 1+R , we get the conclusion. Proposition 2.9. Let D be an H-domain and S ⊂ H. Suppose that there are z0 ∈ R and r > 0 such that {|z − z0 | = r} ∩ D has a connected component Cr , which disconnect S from ∞. In other words, S lies in the bounded component of D \ Cr . Let g be a conformal map from D onto H such that limz→∞ g(z)/z = 1. Then there is w0 ∈ R such that radw0 (g(S)) ≤ 4 radz0 (S). Proof. Since Cr is a crosscut of D, and S lies in the bounded component of D \ Cr , g(Cr ) is a crosscut of g(D) = H, and g(S) lies in the bounded component of H \ g(Cr ). Let w0 be one endpoint of g(Cr ). It suces to show that radw0 (g(Cr )) ≤ 4r. Let L = radz0 (K) and L0 = |z0 − w0 |. Take a big number R > r + L + L′ and let CR = {z ∈ H : |z − z0 | = R}. Then S and CR can be separated by the semi-annulus {z ∈ H : r < |z − x0 | < R} in D. By the comparison principle and conformal invariance of extremal length, 1 dH (g(Cr ), g(CR )) = dD (Cr , CR ) ≥ log(R/r). π 11 Let K be the H-hull H \ D. Then g − gK is a real constant. So we may assume that g = gK . By Proposition 2.3 again, g(C) is a crosscut of H with radw0 (g(C)) ≤ R + L + L0 . Let r′ = radw0 (g(Cr )) and R′ = R + L + L0 . By comparison principle, reection principle and Teichmüller Theorem, dH (g(Cr ), g(CR )) ≤ dH (g(Cr ), {z ∈ H : |z − w0 | = R′ }) = 2dC (g(Cr )doub , {|z − w0 | = R′ }) ≤ 2dC ([−1, 0], {|z| = R′ /r′ } = 2M (R′ /r′ ). By [1, Formula 4-14], 2M (R′ /r′ ) = Λ((R′ /r′ )2 − 1). Thus, by the above displayed formulas and [1, Formula (4-21)] 1 1 1 log(R/r) ≤ Λ((R′ /r′ )2 − 1) ≤ log(16(R′ /r′ )2 ) = (4(R′ /r′ )). π 2π π So we get r′ ≤ 4(R′ /R)r. Letting R → ∞, we get R′ /R → 1. So r′ ≤ 4r. 2.6 Two-sided Chordal SLE Suppose γ is a chordal SLEκ curve in H from 0 to ∞, which generates the ltration F = (Ft )t≥0 . Let P denote the law of γ, and E denote the corresponding expectation. Let √ z ∈ R \ {0}. By (2.4) and the fact that Ut = κBt for some standard Brownian motion Bt , up to τz∗ , Zt (z) and gt′ (z) satisfy the following SDE and ODE: √ 2 dZt (z) = − κdBt + dt; Zt (z) dgt′ (z) −2 ′ = dt. gt (z) Zt (z)2 By Itô's formula (cf. [18]), we get the following continuous positive local martingale: |gt′ (z)|α |z|α Mt (z) := , 0 ≤ t < τz∗ , (2.5) |Zt (z)|α which satises the SDE: dMt (z) κ − 8 dBt = √ . (2.6) Mt (z) κ Zt (z) 12 By Girsanov Theorem (cf. [18]), if we tilt the law P by the local martingale M· (z), we get a new random curve γ e, whose driving function U e satises the SDE: √ κ−8 dU et = κdBet + dt, Zet (z) where B e is another standard Brownian motion, and Zet 's are the centered Loewner maps associated with γ e. In fact, such γ e is a chordal SLEκ (κ − 8) curve (cf. [10]) in H started from κ 0, aimed at ∞, with the force point located at z. Since κ−8 < 2 − 4, with probability 1, γ e ends at z (cf. [15]). The above curve γ e from 0 to z is the rst arm of a two-sided chordal SLEκ curve in H from 0 to ∞ passing through z. Given this arm γ e, the rest of the two-sided chordal SLEκ curve is a chordal SLEκ curve from z to ∞ in the unbounded connected component of H \ γ e. We use P∗z to denote the law of such a two-sided chordal SLEκ curve, and let E∗z denote the corresponding expectation. For r > 0, we use Prz to denote the conditional law P[·|τrz < ∞], i.e., the law of a chordal SLEκ curve in H from 0 to ∞ conditioned to visit the disk with radius r centered at z; and let Erz denote the corresponding expectation. Proposition 2.10. Let z ∈ R \ {0} and R ∈ (0, |z|). Then P∗z is absolutely continuous w.r.t. PRz on FτR R < ∞}, and the Radon-Nikodym derivative is uniformly bounded by some z ∩ {τ z constant Cκ ∈ [1, ∞) depending only on κ. Proof. By symmetry we may assume z > 0. Let τ = τRz . By the construction of P∗z (through tilting P by M· (z)), we have dP∗z |Fτ ∩ {τ < ∞} = Mτ (z). dP|Fτ ∩ {τ < ∞} By the denition of PRz, dPR z |Fτ ∩ {τ < ∞} 1 = . dP|Fτ ∩ {τ < ∞} P[τ < ∞] Thus, it suces to prove that Mτ (z) · P[τ < ∞] is uniformly bounded. By (3.6), P[τ < ∞] ≲ (R/|z|)α . Since gτ maps the simply connected domain Ω := C \ (Kτdoub ∪ (−∞, 0]) 13 conformally onto C \ (−∞, bKτ ], by Koebe's 1/4 theorem, |gτ′ (z)|·R = |gτ′ (z)|·dist(z, ∂Ω) ≍ dist(gτ (z), ∂(C\(−∞, bKτ ])) = |gτ (z)−bKτ | ≤ Zt (z), (2.7) where in the last step we used gτ (z) > bKτ ≥ Uτ . Thus, |gt′ (z)|α |z|α Rα |gt′ (z)|α R|α Mτ (z) · P[τ < ∞] ≲ = ≲ 1. |Zt (z)|α |z|α |Zt (z)|α Proposition 2.11. Let z ∈ R \ {0} and 0 < r < η < |z|. Then Prz restricted to Fτηz is absolutely continuous with respect to P∗z , and there is a constant β > 0 depending only on κ such that  dPr |Fτ z   r β if r/η < 1/6. z η log ∗ ≲ , (2.8) dPz |Fτηz η Proof. Recall the G(z) = b c|z|−α dened by (1.6). Dene Gt (z) = |Zt′ (z)|α Zt (z) if τz∗ > t; and Gt (z) = 0 if τz∗ ≤ t. Then dP∗z |Fτηz Gτ z (z) = Mτηz (z) = η . dP|Fτηz G(z) By the denition of Prz , we have dPrz |Fτηz P[τrz < ∞|Fτzη ] = . dP|Fτηz P[τrz < ∞] Since P[τrz < ∞|Fτzη ] = 0 implies that τzη ≤ τz∗ , which in turn implies that Gτηz (z) = 0, by the above two displayed formulas, Prz restricted to Fτηz is absolutely continuous with respect to P∗z , and dPrz |Fτηz P[τrz < ∞|Fτzη ]/(Gτηz (z)rα ) = . dP∗z |Fτηz P[τrz < ∞]/(G(z)rα ) By (1.7) and Koebe's distortion theorem, there are constants β, δ > 0 such that, if r/η < 1/6, then log(P[τrz < ∞]/(G(z)rα )) ≲ (r/|z|)β , log(P[τrz < ∞|Fτzη ]/(Gτηz (z)rα )) ≲ (r/η)β . The above two displayed formulas together imply (2.8). 14 CHAPTER 3 MAIN ESTIMATES In this chapter, we will provide some useful estimates for the proof of the main theorem. We use the notion and symbols in the previous chapter. We now dene the function F (z1 , . . . , zn ) that appeared in Theorem 1.1. From now on, let d0 = 1 + κ 8 and α = κ8 − 1. For y ≥ 0, dene Py on [0, ∞) by   y α−(2−d0 ) x2−d0 , x ≤ y;  Py (x) =  xα ,  x ≥ y. For an (ordered) set of distinct points z1 , . . . , zn ∈ H \ {0}, we let z0 = 0 and dene yk = Im zk , lk = min {|zk − zj |}, Rk = min {|zk − zj |}, 1 ≤ k ≤ n. (3.1) 0≤j≤k−1 0≤j≤n,j̸=k Note that we have Rk ≤ lk . For r1 , . . . , rn > 0, dene n Y Pyk (rk ) F (z1 , . . . , zn ; r1 , . . . , rn ) = . (3.2) k=1 P yk (lk ) The following is [16, Formula (2.7)]. For any permutation σ of {1, . . . , n}, F (z1 , . . . , zn ; r1 , . . . , rn ) ≍ F (zσ(1) , . . . , zσ(n) ; rσ(1) , . . . , rσ(n) ). (3.3) The following proposition combines [17, Theorem 1.1] (which gives the upper bound) and [16, Theorem 4.3] (which gives the lower bound). Proposition 3.1. Let z1 , . . . , zn be distinct points on H \ {0}. Let R1 , . . . , Rn be dened by (3.1). Let rj > 0, 1 ≤ j ≤ n. Then for a chordal SLEκ curve γ in H from 0 to ∞, we have Pyj (rj ) z ˆ P[τrjj < ∞, 1 ≤ j ≤ n] ≲ ; Qn j=1 (1 ∧ Pyj (lj ) ) ˆ P[τrjj < ∞, 1 ≤ j ≤ n] ≳ F (z1 , . . . , zn ; r1 , . . . rn ), if rj ≤ Rj , 1 ≤ j ≤ n. z 15 Now suppose z1 , . . . , zn are distinct points on R \ {0}. Then yk = 0, 1 ≤ k ≤ n. So, the Pyk (rk ) rkα in (3.2) simplies to α . Then we dene Pyk (lk ) lk Yn Yn F (z1 , . . . , zn ) = rk−α F (z1 , . . . , zn ; r1 , . . . , rn ) = lk−α . (3.4) k=1 k=1 This function is dierent from the F (z1 , . . . , zn ) that appeared in [16], which was dened for z1 , . . . , zn ∈ H. By (3.3), we have F (z1 , . . . , zn ) ≍ F (zσ(1) , . . . , zσ(n) ). (3.5) A simple but useful special case of Proposition 3.1 is: when n=1 and z1 ∈ R \ {0}, we have P[τrz11 < ∞] ≍ (r1 /|z1 |)α , 0 < r < |z1 |. (3.6) The estimate includes a lower bound and an upper bound. They rst appeared in [2]. The upper bound in (3.6) was called the boundary estimate in the literature. From now on till the end of this chapter, P denotes the law of a chordal SLEκ curve in H from 0 to ∞; for z ∈ R \ {0} and r > 0, Prz denotes the conditional law P[·|τrz < ∞], and P∗z denotes the law of a two-sided chordal SLEκ curve in H from 0 to ∞ passing through z. When γ follows some law above in the context, let U t , Kt and gt be respectively the chordal Loewner driving function, hulls and maps which correspond to γ. Let Zt = gt − Ut be the centered Loewner maps, and let Ht = H \ Kt . For t ≥ 0, let St+ be the set of prime ends of Ht that lie on the right side of γ[0, t] or on [bKt , ∞), and let St− be the set of prime ends of Ht that lie on the left side of γ[0, t] or on (−∞, aKt ]. More precisely, St+ and St− are respectively the images of [0, +∞) and (−∞, 0] under Zt−1 . Proposition 3.2. Let z1 , . . . , zn be distinct points in R \ {0}, where n ≥ 2. Let R1 , . . . , Rn be dened by (3.1). Let rj ∈ (0, Rj /8), 1 ≤ j ≤ n. Then we have a constant β > 0 such that 16 for any k0 ∈ {2, . . . , n} and s0 ≥ 0, P[τrz11 < τrzkk < ∞, 2 ≤ k ≤ n; dist(zk0 , γ[0, τrz11 ]) ≤ s0 ] n Y  s0 β ≲F (z1 , . . . , zn ) rjα . j=1 |zk0 − z1 | ∧ |zk0 | Proof. This proposition is very similar to [16, Theorem 3.1]. The following estimate is [16, Formula (A.14)]. For distinct points z1 , . . . , zn ∈ H\{0}, rj ∈ (0, Rj ), 1 ≤ j ≤ n, and s0 > 0,  s0  α2 32n P[τrzjj < ∞, 1 ≤ j ≤ n; τsz01 < τrz22 < τrz11 ] ≲ F (z1 , . . . , zn ; r1 , . . . , rn ) · . |z1 − z2 | ∧ |z1 | α Let k0 ∈ {2, . . . , n} and β= 32n2 . Applying (3.3) to the above formula with a permutation σ of Nn , which sends 1 to k0 and 2 to 1, we nd that z z  s0 β P[τrzjj < ∞, 1 ≤ j ≤ n; τs0k0 < τrz21 < τs0k0 ] ≲ F (z1 , . . . , zn ; r1 , . . . , rn ) · . |zk0 − z1 | ∧ |zk0 | We then complete the proof by setting z1 , . . . , zn ∈ R \ {0}. Proposition 3.3. Let z1 ∈ R\{0} and 0 ≤ s < r < R∧|z1 |. On the event {τrz1 < τz∗1 }, let ξ+ be the connected component of {|z − z1 | = R} ∩ Hτrz1 with one endpoint being z1 + sign(z1 )R; otherwise let ξ+ = ∅. Let Er,s;R = {γ[τrz1 , τsz1 ] ∩ ξ+ = ∅}. Then (i) If s > 0, Psz1 [Er,s;R c ] ≲ (r/R)α . (ii) If s = 0, P∗z1 [Er,0;R c ] ≲ (r/R)α . Proof. (i) Assume that z1 > 0 by the left-right symmetry of chordal SLE. Suppose γ follows the law P. Since κ ∈ (0, 8), the probability that γ visits {z1 + s, z1 − s, z1 + R} is zero. We now assume that γ does not visit this set. Let τ = inf({t ≥ τrz1 : γ(t) ∈ ξ+ } ∪ {∞}). Then τ is a stopping time, and Er,s;R = {τ < τsz1 < ∞}. Let Eτ = {τ < τsz1 } ∈ Fτ . By DMP 17 of chordal SLE, conditionally on Fτ and the event Eτ , there is a random curve γ e following the law P such that γ(τ + ·) = Zτ−1 ◦ γ e. Let D = {z ∈ H \ Kτ : |z − z1 | ≤ s}, D e = Zτ (D), ze1 = Zτ (z1 ) > 0, and se = radze1 (D) e > 0. On the event Eτ , in order for Er,s;R to happen, we need that γ(τ + ·) visits D, which is equivalent to that γ e visits De. By (3.6), c P[Er,s;R |Fτ , Eτ ] ≲ (1 ∧ (e z1 ))α . r1 /e By Lemma 2.8 and conformal invariance of extremal length, − 1 ∧ (er1 /ez1 ) ≲ e−πdH ((−∞,0],D) = e−πdHτ (Sτ ,D) . e Since Sτ− can be separated from D in Hτ by the semi-annulus {s < |z − z1 | < R}, by the comparison principle of extremal length, dHτ (Sτ− , D) ≥ log(R/s)/π . So by the above two displayed formulas we get c P[Er,s;R |Fτ , Eτ ] ≲ (s/R)α , which together with P[Eτ ] ≤ P[τrz1 < ∞] ≲ (r/|z1 |)α (the upper bound in (3.6)) implies that c P[Er,s;R ] ≲ (r/|z1 |)α (s/R)α . Combining this estimate with the lower bound in (3.6), i.e., P[τsz1 ] ≳ (s/z1 )α , we get (i). (ii) From Proposition 2.10 and (i), we get P∗z1 [Er,s;R c ] ≲ (r/R)α for any s ∈ (0, r). We then complete the proof by sending s to 0+ . Lemma 3.4. Let z1 , . . . , zn , w1 , . . . , wm be distinct points in R \ {0}, where n ≥ 1 and m ≥ 0. Suppose that all zj have the same sign σz ∈ {+, −}, all wk have the same sign σw ∈ {+, −}, σz ̸= σw , and both j 7→ |zj | and k 7→ |wk | are increasing. Let z0 = w0 = 0, zn+1 = σz · ∞, and wm+1 = σw · ∞. Let rj ∈ (0, (|zj − zj−1 | ∧ |zj − zj+1 |)/2), 1 ≤ j ≤ n, and sk ∈ (0, (|wk − wk+1 | ∧ |wk − wk−1 |)/2), 1 ≤ k ≤ m. Let R > 2(|zn | ∨ |wm |). Then P[τR∞ < τrzjj < ∞, 1 ≤ j ≤ n; τR∞ < τswkk < ∞, 1 ≤ k ≤ m]  |z | α Y n Ym 1 α ≲ F (z1 , . . . , zn , w1 , . . . , wm ) rj · sαk . (3.7) R j=1 k=1 Proof. By symmetry, we may assume that wm < · · · < w1 < 0 < z1 < · · · < zn . Dene Fz Qn |zj − zj−1 |−α ; Fw = m −α Q and Fw such that Fz = j=1 k=1 |wk − wk−1 | , if m ≥ 1; and Fw = 1 if m = 0. Then we have F (z1 , . . . , zn , w1 , . . . , wm ) = Fz Fw . 18 Let τ = τR∞ . Let E denote the event in (3.7). Then E = E∗τ ∩ E# , where E∗τ := {τR∞ < τrzjj ∧ τz∗j , 1 ≤ j ≤ n; τR∞ < τswkk ∧ τw∗ k , 1 ≤ k ≤ m} ∈ Fτ ; E# : = {τrzjj < ∞, 1 ≤ j ≤ n; τswkk < ∞, 1 ≤ k ≤ m}. Suppose the event E∗τ occurs. Let zej = Zτ (zj ), Dj = {z ∈ Hτ : |z − zj | ≤ rj }, D ej = Zτ (Dj ), and rej = radzej (D e j ), 1 ≤ j ≤ n. Let wek = Zτ (wk ), Ek = {z ∈ Hτ : |z − wk | ≤ sk }, Eek = Zτ (Ek ), and sek = radwek (E ek ), 1 ≤ k ≤ m. Then wem < · · · < w e1 < 0 < ze1 < · · · < zen . By DMP of chordal SLEκ and Proposition 3.1, n−1  ren α Y  rej α P[E# |Fτ , E∗τ ] ≲ 1∧ · 1∧ |e zn | j=1 |e zj | ∧ |e zj − zej+1 | m−1  sem α Y  sek α · 1∧ · 1∧ . (3.8) |w em | k=1 |wek | ∧ |w ek − w ek+1 | Here we organize zej 's and wek 's by zen , . . . , ze1 , wem , . . . , w e1 when applying Proposition 3.1. In the case that m = 0, the second line disappears. By Proposition 2.8 and conformal invariance of extremal distance, rej − 1∧ ≲ e−πdH ((−∞,0],Dj ) = e−πdHτ (Sτ ,Dj ) , 1 ≤ j ≤ n; (3.9) e |e zj | rej 1∧ ≲ e−πdH ([ezj+1 ,∞),Dj ) = e−πdHτ ([zj+1 ,∞),Dj ) , 1 ≤ j ≤ n − 1; (3.10) e |ezj − zej+1 | sek + 1∧ ≲ e−πdH ([0,+∞),Ek ) = e−πdHτ (Sτ ,Ek ) , 1 ≤ k ≤ m; (3.11) e |wek | sek 1∧ ≲ e−πdH ((−∞,wek+1 ],Ek ) = e−πdHτ ((−∞,wk+1 ],Ek ) , 1 ≤ k ≤ m − 1. (3.12) e |w ek − w ek+1 | Since Sτ− can be separated from Dj in Hτ by {z ∈ H : rj < |z − zj | < R − |zj |}, by comparison principle of extremal distance, 1  R − |z |  j dHτ (Sτ− , Dj ) ≥ log , π rj which combined with (3.9) and that R > 2|zj | implies that rej rj rj 1∧ ≲ ≍ , 1 ≤ j ≤ n. (3.13) |e zj | R − |zj | R 19 Figure 3.1: A gure for the proof of Lemma 3.4. This gure illustrates an application of the comparison principle of extremal distance in the proof of Lemma 3.4. Here n = 3 and m = 2. The curve γ is stopped at the time τ = τR∞ . Assume that the event E∗τ occurs. To − bound the extremal distance dHτ (D2 , Sτ ) from below for example, we use the semi-annulus (shaded region) A2 := {z ∈ H : r2 < |z − z2 | < R − |z2 |} and the fact that any curve in Hτ that connects the semi-circle ∂D2 ∩ H with the left side of γ[0, τ ] or the real interval (−∞, 0] must cross A2 , i.e., contain a subpath in A2 connecting its two semi-circles. The intersection of A2 with γ[0, τ ] does not cause a problem in the application. See Figure 3.1. Since [zj+1 , ∞) can be separated from Dj in Hτ by {z ∈ H : rj < |z − zj | < |zj+1 − zj |}, by comparison principle of extremal distance, 1  |z − z |  j+1 j dHτ ([zj+1 , ∞), Dj ) ≥ log , π rj which combined with (3.10) implies that rej rj 1∧ ≲ , 1 ≤ j ≤ n − 1. (3.14) |e zj − zej+1 | |zj+1 − zj | For 1 ≤ j ≤ n − 1, since R − |zj | ≥ |zj+1 | − |zj | = |zj+1 − zj |, by (3.13) and (3.14), rej rj 1∧ ≲ , 1 ≤ j ≤ n − 1. (3.15) |e zj | ∧ |e zj − zej+1 | |zj+1 − zj | Similarly, sek sk sk 1∧ ≲ ≍ , 1 ≤ k ≤ m; (3.16) |wek | R − |wk | R 20 sek sk 1∧ ≲ , 1 ≤ k ≤ m − 1. (3.17) |wek | ∧ |w ek − w ek+1 | |wk+1 − wk | Combining (3.8) with (3.13) (for j = n), (3.15), (3.16) (for k = m) and (3.17), we get  r α n−1Y rj α  s α m−1 Y sk α n m P[E# |Fτ , E∗τ ] ≲ · R j=1 |zj − zj+1 | R k=1 |wk − wk+1 |  |z | α Yn Y m 1 α ≤ Fz Fw rj sαk R j=1 k=1 Here, if m = 0, the factors involving sk and sm disappear; if m ≥ 1, we used that R ≥ |w1 | in the estimate. Since F (z1 , . . . , zn , w1 , . . . , wm ) = Fz Fw , taking expectation we get (3.7). Lemma 3.5. Suppose x0 , . . . , xN , N ≥ 1, are distinct points in R \ {0} that have the same sign ν ∈ {+, −}, and j 7→ |xj | is increasing. Let xN +1 = ν · ∞. Let Rj = (|xj − xj+1 | ∧ |xj − xj−1 |)/2 and rj ∈ (0, Rj ), 1 ≤ j ≤ N . Let r0 ∈ (0, |x0 − x1 |/2). Then  r α NY −1  N  N rk α Y rk α P[τrxjj < τrx00 < ∞; 1 ≤ j ≤ N ] ≲ · (3.18) |xN | k=0 |xk − xk+1 | k=1 |xk − xk−1 |  R α  N  N  N r0 α Y rk 2α  r0 α Y rk 2α ≲ ≤ . (3.19) |xN | |x0 − x1 | k=1 Rk |x0 − x1 | k=1 Rk Proof. Assume all xj 's are positive by symmetry. Let P denote the RHS of (3.18) (depending x ∗ on x0 , . . . , x N and r0 , . . . , rN ). We write τj for τrjj , 1 ≤ j ≤ N . Let SN denote the set of ∗ permutation σ of {0, 1, . . . , N } such that σ(n) = 0. For each σ ∈ SN , let Eσ = {τσ(0) < S τσ(2) < · · · < τσ(N ) < ∞}. Then σ∈SN∗ Eσ is the event in (3.18). To prove (3.18), it suces ∗ to show that, for any σ ∈ SN , P[Eσ ] ≲ P . ∗ Fix σ ∈ SN . For 0 ≤ k ≤ N − 1, let Ekσ = {τσ(0) < τσ(1) < · · · < τσ(k) < τσ(k+1) ∧ τx∗σ(k+1) } ∈ Fτσ(k) ; σ and let EN = Eσ . Then E0σ ⊃ E1σ ⊃ · · · ⊃ EN σ = Eσ . Let Sσ = {j : n − 1 ≥ j ≥ σ −1 (n), σ(j + 1) < σ(j)}. (3.20) 21 Figure 3.2: The rst gure for the proof of Lemma 3.5. This gure illustrates a situation in the proof of Lemma 3.5. Here n = 5, and the event Eσ happens, where σ = 02 15 23 34 41 50 . We have Sσ = {1, 3, 4} since σ −1 (5) = 1, σ(1) = 5 > 3 = σ(2),  σ(3) = 4 > 1 = σ(4), σ(4) = 1 > 0 = σ(5), but σ(2) = 3 < 4 = σ(3). We have S1σ = ∅ −1 because the only index between σ(2) and σ(1) is 4, and σ (4) = 3 > 2. We have S3 = {2, 3} because 2, 3 lie between σ(4) and σ(3), and σ (2), σ −2 (3) < 3. We have σ −1 S4σ = ∅ because there is no index that lies between σ(5) and σ(4). For each j ∈ Sσ , let Sjσ = {k : σ(j + 1) < k < σ(j), σ −1 (k) < j}. (3.21) In plain words, Sσ is the set of index j ≥ j0 , where j0 := σ −1 (N ), such that σ(j + 1) < σ(j); and Sjσ is the set of index k, which lies strictly between σ(j + 1) and σ(j), such that the disc {|z − xk | ≤ rk } was visited by γ before {|z − xσ(j) | ≤ rσ(j) }. For example, j0 and N −1 belong to Sσ. For j ∈ Sσ , the set Sjσ may or may not be empty. See Figure 3.2. 2r 2r For 0 ≤ j ≤ N − 1, let Q+ j α j = ( |xj −xj+1 | ) . For 1 ≤ j ≤ N , let Q− j α j = ( |xj −xj−1 | ) . Let Q −1 + QN Qn = ( |xrNN | )α . Then P = QN · N j=1 Qj · − j=2 Qj . By (3.6), P[Eσσ−1 (N ) ] ≤ P[τN < ∞] ≲ QN . (3.22) We claim that, for any j ∈ Sσ , Y − σ P[Ej+1 |Fτσ(j) , Ejσ ] ≲ Q− + σ(j) Qσ(j+1) (Q+k Qk ); (3.23) k∈Sjσ and Y Y  N Y −1 YN Q− + σ(j) Qσ(j+1) (Q+ − k Qk ) ≤ Q+l · Q−l . (3.24) j∈Sσ k∈Sjσ l=0 l=1 22 Note that (3.22,3.23,3.24) together imply that P[Eσ ] = P[Enσ ] ≲ P . We rst prove (3.24). It suces to show that [ {0, . . . , N − 1} ⊂ ({σ(j + 1)} ∪ Sjσ ); (3.25) j∈Sσ [ {1, . . . , N } ⊂ ({σ(j)} ∪ Sjσ ). (3.26) j∈Sσ Let l ∈ {0, . . . , N − 1}. We consider several cases. Case 1. σ −1 (l) < σ −1 (N ). Since σ −1 (0) = N , we have l ≥ 1. Since σ(σ −1 (N )) = N > l and σ(N ) = 0 < l, there exists σ −1 (N ) ≤ j0 ≤ N − 1 such that σ(j0 ) > l > σ(j0 + 1). By (3.20,3.21) we have j0 ∈ Sσ and l ∈ Sjσ0 . Case 2. σ −1 (l) ≥ σ −1 (N ). Then σ −1 (l) − 1 ≥ σ −1 (N ) since l ̸= N . Consider two subcases. Case 2.1. σ(σ −1 (l) − 1) > σ(σ −1 (l)) = l. In this subcase, j1 := σ −1 (l) − 1 ∈ Sσ by (3.20), and σ(j1 + 1) = l. Case 2.2. σ(σ −1 (l) − 1) < σ(σ −1 (l)) = l. Since σ(σ −1 (N )) = N > l > σ(σ −1 (l) − 1) and σ −1 (N ) ≤ σ −1 (l) − 1, there exists σ −1 (N ) ≤ j2 ≤ σ −1 (l) − 2 such that σ(j2 ) > l > σ(j2 + 1). This implies that j2 ∈ Sσ and l ∈ Sjσ2 . Thus, in all cases, there is some j ∈ Sσ such that l ∈ {σ(j + 1)} ∪ Sjσ . So we get (3.25). Let l ∈ {1, . . . , N }. We consider several cases. Case 1. σ −1 (l) < σ −1 (N ). Then l ≤ N − 1. By Case 1 of the last paragraph, there exists j0 ∈ Sσ such that l ∈ Sjσ0 . Case 2. σ −1 (l) ≥ σ −1 (N ). Consider two subcases. Case 2.1. σ(σ −1 (l)) > σ(σ −1 (l) + 1). In this subcase, j1 := σ −1 (l) ∈ Sσ and l = σ(j1 ). Case 2.2. l = σ(σ −1 (l)) < σ(σ −1 (l) + 1). Since σ(σ −1 (l) + 1) > l > 0 = σ(N ), there exists σ −1 (l) + 1 ≤ j2 ≤ N − 1 such that σ(j2 ) > l > σ(j2 + 1). This implies that j2 ∈ Sσ and l ∈ Sjσ2 . Thus, in all cases, there is some j ∈ Sσ such that l ∈ {σ(j)} ∪ Sjσ . So we get (3.26). Combining (3.25,3.26) we get (3.24). Finally, we prove (3.23). Fix j ∈ Sσ . Let τ = τσ(j) . Suppose the event Ejσ occurs. Let w = xσ(j+1) , D = {z ∈ H : |z − w| ≤ rσ(j+1) }, w e = Zτ (w), D e = Zτ (D) e , and re = radwe (D)e . By DMP of chordal SLEκ and (3.6), σ  re α P[Ej+1 |Fτ , Ejσ ] ≤ P[τσ(j+1) < ∞|Fτ , Ejσ ] ≲ 1 ∧ . (3.27) |w| e 23 Figure 3.3: The second gure for the proof of Lemma 3.5. This gure illustrates an application of the comparison principle of extremal distance in the proof of Lemma 3.5.  Here n = 4, and the event Eσ happens, where σ = 0 1 2 3 4 3 4 1 2 0 . We stop the curve at the time τ := τ4 . Then the next semi-disc to visit is D1 = {z ∈ H : |z − x1 | ≤ r1 }. We know σ − that 1 ∈ Sσ , σ(1) = 4, σ(2) = 1, and S1 = {3}. The D1 is separated from Sτ in Hτ by the + − e+ disjoint regions A1 , A4 , A e3 and A e3 , among which A1 and A4 are semi-annuli, and A 3 and Ae− 3 are subsets of two semi-annuli, which have the same center x 3 , same inner radius r3 , but dierent outer radii. By Proposition 2.8 and conformal invariance of extremal distance, re − 1∧ ≲ e−πdH ((−∞,0],D) = e−πdHτ (Sτ ,D) . (3.28) e |w| e Dene semi-annuli Aσ(j) = {z ∈ H : rσ(j) < |z − xσ(j) | < |xσ(j)−1 − xσ(j) |/2}; Aσ(j+1) = {z ∈ H : rσ(j+1) < |z − xσ(j+1) | < |xσ(j+1)+1 − xσ(j+1) |/2}; A± k = {z ∈ H : rk < |z − xk | < |xk±1 − xk |/2}, k ∈ Sjσ . k ∈ Sjσ , e± A± For each dene A k to be the connected component of k ∩ Hτ whose boundary contains xk ± r k . Then e+ Aσ(j) , Aσ(j+1) , A and e− , k ∈ S σ , A are mutually disjoint. See Figure k k j 3.3. Since the event Ejσ occurs, any curve in Hτ connecting D with Sτ− must contain a subarc crossing Aσ(j) , a subarc crossing Aσ(j+1) , a subarc contained in e+ A crossing A+ for k k each k ∈ Sjσ , and a subarc contained in Ae− crossing A− for each k ∈ Sjσ . By the comparison k k 24 principle and composition rule of extremal length, we know that  |x σ(j) − xσ(j)−1 |  |x 1 σ(j+1) − xσ(j+1)+1 |  1  dHτ (Sτ− , D) ≥ log + log π 2rσ(j) π 2rσ(j+1) 1 X  |xk − xk+1 |  1 X  |xk − xk−1 |  + log + log . (3.29) π k∈S σ 2rk π k∈S σ 2rk j j Combining (3.27,3.28,3.29) we get (3.23). Then we get (3.18), which implies (3.19) because |xk − xk±1 | ≥ Rk , 1 ≤ k ≤ N , and RN = |xN − xN −1 | ≤ |xN |. Remark 3.6. By a slight modication of the above proof, we can obtain the following estimate. Let x0 , . . . , xN +1 , R1 , . . . , RN , r0 , . . . , rN be as in Lemma 3.5. Let I = [a, x0 ] for some a ∈ (0, x0 ), and τrI0 = τI×[0,r0 ] . Then N  Y rj 2α P[τrzjj < τrI0 < ∞; 1 ≤ j ≤ N ] ≲ . j=1 Rj To prove the estimate, we may use the same extremal length argument except that we do not use a semi-annulus centered at x0 because such a semi-annulus may not disconnect I × [0, r0 ] from other xj 's in Hτ . So we have the same factor in the upper bound except for ( |x1r−x 0 0| )α . Lemma 3.7. Let zj , 0 ≤ j ≤ n + 1, wk , 0 ≤ k ≤ m + 1, rj , 1 ≤ j ≤ n, sk , 1 ≤ k ≤ m, be as in Lemma 3.4. Now assume n ≥ 2. Let j0 ∈ {2, . . . , n}. Let j0 −1 m−1 Y Y −α −α −α Q =|zj0 −1 − zj0 | · |zj − zj+1 | · |wm | · (|wk | ∧ |wk − wk+1 |)−α j=1 k=1 Yn · (|zj0 | ∧ |zj0 − zj0 +1 |)−α · (|zj − zj−1 | ∧ |zj − zj+1 |)−α . j=j0 +1 Here when m = 0, the |wm |−α · (|wk | ∧ |wk − wk+1 |)−α disappears; and when j0 = n, Qm−1 k=1 the − zj−1 | ∧ |zj − zj+1 |)−α disappears. Then we have Qn j=j0 +1 (|zj n m z z Y Y P[τrjj00 < τrzjj < ∞, j ∈ Nn \ {j0 }; τrjj00 < τswkk < ∞, k ∈ Nm ] ≲ Qrjα0 · rjα · sαk . (3.30) j=1 k=1 25 Figure 3.4: A gure for the proof of Lemma 3.7. This gure illustrates the event E in Lemma 3.7. Here n = 3, m = 2, and j0 = 2. The curve γ visits the ve semi-discs centered z at z1 , z2 , z3 , w1 , w2 , among which the one centered at z2 is rst visited (at the time τ = τr 2 ). 2 The parts of γ before τ and after τ are respectively drawn in solid and dashed lines. z Proof. By symmetry, we may assume that wm < · · · < w1 < 0 < z1 < · · · < zn . Let τ = τrjj00 . Let E denote the event in (3.30). See Figure 3.4. Let E∗τ = {τ < τrzjj ∧ τz∗j : j ∈ Nn \ {j0 }} ∈ Fτ ; E# = {τrzjj < ∞, j ∈ Nn \ {j0 }; τswkk < ∞, 1 ≤ k ≤ m}. Then E = E∗τ ∩ E# . By (3.6), P[E∗τ ] ≲ (rj0 /|zj0 |)α . Suppose the event E∗τ occurs. Let zej = Zτ (zj ), Dj = {z ∈ Hτ : |z − zj | ≤ rj }, D ej = Zτ (Dj ), and rej = radzen (D e n ), 1 ≤ j ≤ n. Let wek = Zτ (wk ), Ek = {z ∈ Hτ : |z − wk | ≤ sk }, E ek = Zτ (Ek ), and sek = radwem (E em ), 1 ≤ k ≤ m. Then wem < · · · < w e1 < 0 < ze1 < · · · < zen . By DMP of chordal SLEκ and Proposition 3.1 and that E = E∗τ ∩ E# , j0 −1  n Y rej α Y  rej α P[E|Fτ , E∗τ ] ≲ 1∧ · 1∧ j=1 |ezj | ∧ |e zj − zej+1 | j=j +1 |e zj − zej−1 | 0 m−1  sem α Y sek α · 1∧ · 1∧ . (3.31) |w em | k=1 |w ek | ∧ |wek − w ek+1 | Here when applying Proposition 3.1, we ordered the points zej and wek by zej0 , . . . , ze1 , zej0 +1 , . . . , zen , w em , . . . , we1 , 26 rej0 α and omit the factor (1 ∧ |e zj0 | ) , which is bounded by 1. By Proposition 2.8 and conformal invariance of extremal length, rej − 1∧ ≲ e−πdH ((−∞,0],Dj ) = e−πdHτ (Sτ ,Dj ) , 1 ≤ j ≤ j0 − 1; (3.32) e |e zj | sek + 1∧ ≲ e−πdH ([0,+∞),Ek ) = e−πdHτ (Sτ ,Ek ) , 1 ≤ k ≤ m; (3.33) e |w ek | rej 1∧ ≲ e−πdH ([ezj+1 ,∞),Dj ) = e−πdHτ ([zj+1 ,∞),Dj ) , 1 ≤ j ≤ j0 − 1; (3.34) e |e zj − zej+1 | rej 1∧ ≲ e−πdH ((−∞,ezj−1 ],Dj ) = e−πdHτ (Kτ ∪(−∞,zj−1 ],Dj ) , j0 + 1 ≤ j ≤ n; (3.35) e |e zj − zej−1 | sek 1∧ ≲ e−πdH ((−∞,wek+1 ],Ek ) = e−πdHτ ((−∞,wk+1 ],Ek ) , 1 ≤ k ≤ m − 1. (3.36) e |w ek − w ek+1 | Since Sτ+ and Em are separated by the semi-annulus {z ∈ H : sm < |z − wm | < |wm |} in Hτ , we have dHτ (Sτ+ , Em ) ≥ 1 π log( |wsmm | ), which together with (3.33) implies that sem sm 1∧ ≲ . (3.37) |w em | |wm | For 1 ≤ j ≤ j0 − 2, since Dj is separated from both Sτ− and [zj+1 , ∞) by the semi-annulus {z ∈ H : rj < |z − zj | < |zj − zj+1 |} in Hτ , we have 1  |z − z |  j j+1 dHτ (Sτ− , Dj ), dHτ ([zj+1 , +∞), Dj ) ≥ log , π rj which combined with (3.32,3.34) implies that rej  rej   rej  rj 1∧ = 1∧ ∨ 1∧ ≲ . (3.38) |e zj | ∧ |ezj − zej+1 | |e zj | |e zj − zej+1 | |zj − zj+1 | For j = j0 − 1, we have a better estimate. Since Dj0 −1 is separated from both Sτ− and [zj0 , ∞) by a disjoint pair of semi-annuli {z ∈ H : rj0 −1 < |z − zj0 −1 | < |zj0 −1 − zj0 |/2} and {z ∈ H : rj0 < |z − zj0 | < |zj0 −1 − zj0 |/2}, we have  |z j0 −1 − zj0 |  |z 1 j0 −1 − zj0 |  1  dHτ (Sτ− , Dj0 −1 ), dHτ ([zj0 , +∞), Dj0 −1 ) ≥ log + log ) , π 2rj0 −1 π 2rj0 which combined with (3.32,3.34) implies that rej0 −1 rj0 −1 rj0 1∧ ≲ · . (3.39) |e zj0 −1 | ∧ |e zj0 −1 − zej0 | |zj0 −1 − zj0 | |zj0 −1 − zj0 | 27 For 1 ≤ k ≤ m − 1, since Ek is separated from Sτ+ by {z ∈ H : sk < |z − wk | < |wk |} in Hτ , we get dHτ (Sτ+ , Ek ) ≥ π 1 log( |w k| 2sk ). Since Ek is separated from (−∞, wk+1 ] by {z ∈ H : sk < |z − wk | < |wk − wk+1 |} in Hτ , we get dHτ ((−∞, wk+1 ], Ek ) ≥ 1 π log( |wk −w 2sk k+1 | ). These two lower bounds of extremal lengths combined with (3.33,3.36) imply that sek sk 1∧ ≲ . (3.40) |wek | ∧ |w ek − w ek+1 | |wk − wk+1 | ∧ |wk | Suppose j0 = n. Combining (3.31,3.37-3.40), we get α  s α jY 0 −1  α m−1  rj0 m rj Y sk α P[E|Fτ , E∗τ ] ≲ · · , |zj0 −1 − zj0 | |wm | j=1 |z j − zj+1 | k=1 |w k − w k+1 | ∧ |w k | which together with P[E∗τ ] ≲ (rj0 /|zj0 |)α and zj0 +1 = ∞ implies (3.30) for j0 = n. Now suppose 2 ≤ j0 ≤ n − 1. Let N(j0 ,n] = {j0 + 1, . . . , n}. For j ∈ N(j0 ,n] , let Rj = (|zj −zj−1 |∧|zj −zj+1 |)/2. For each k = (kj0 +1 , . . . , kn ) ∈ (N∪{0})N(j0 ,n] , let Sk = {j ∈ N(j0 ,n] : Kj ≥ 1}, and Ek denote the event that τ <∞ and dist(zj , Kτ ) ≥ Rj , for j ∈ N(j0 ,n] \ Sk , and Rj e−kj ≤ dist(zj , Kτ ) < Rj e1−kj for j ∈ Sk . We now bound P[Ek ]. If Sk = ∅, we use (3.6) to conclude that z P[Ek ] ≤ P[τr0j0 < ∞] ≲ (rj0 /|zj0 |)α . Suppose Sk ̸= ∅. We express Sk = {j1 < · · · < jN }. Let xs = zjs , 0 ≤ s ≤ N . By the denition of Ek and Lemma 3.5, we have P[Ek ] ≤ P[τex1−k s js R < τrxj00 < ∞, 1 ≤ s ≤ N ] js N  1−kj n  rj0 α Y e sR js 2α  rj0 α Y ≲ ≲ e−2αkj . |x1 − x0 | s=1 Rjs |zj0 − zj0 +1 | j=j +1 0 Combining the two formulas, we conclude that, for any k ∈ (N ∪ {0})N(j0 ,n] n  rj0 α Y P[Ek ] ≲ e−2αkj . (3.41) |zj0 | ∧ |zj0 − zj0 +1 | j=j +1 0 Suppose for some k = (kj0 +1 , . . . , kn ), Ek ∩ E∗ happens. We claim that rej rj 1∧ ≲ , j0 + 1 ≤ j ≤ n. (3.42) |ezj − zej−1 | Rj e−kj 28 Let j ∈ N(j0 ,n] . First, (3.42) holds trivially if rj ≥ Rj e−kj . Suppose that rj < Rj e−kj . Then Dj can be disconnected from Kτ and (−∞, zj−1 ] in Hτ by {z ∈ H : rj < |z| < Rj e−kj }. By comparison principle of extremal distance, we have 1  R e−kj  j dHτ (Kτ ∪ (−∞, zj−1 ], Dj ) ≥ log , π rj which together with (3.35) implies (3.42). So the claim is proved. Combining (3.31,3.37-3.42) and that Rj = (|zj − zj−1 | ∧ |zj − zj+1 |)/2, we get α  s α m−1 α jY 0 −1   rj0 m Y sk rj α P[E ∩ Ek ] ≲ · · |zj0 − zj0 −1 | |wm | k=1 |wk − wk+1 | ∧ |wk | j=1 |zj − zj+1 | n n  rj0 α Y −α Y · (|zj − zj−1 | ∧ |zj − zj+1 |) · e−αkj . |zj0 | ∧ |zj0 − zj0 +1 | j=j +1 j=j +1 0 0 Summing the inequality over k ∈ (N ∪ {0})N(j0 ,n] , we get (3.30). Lemma 3.8. Let n, m, j0 , z0 , . . . , zbj0 , . . . , zn+1 , and w0 , . . . , wm+1 be as in Lemma 3.7. Here the symbol zbj0 means that zj0 is missing in the list. Let I be a compact real interval that lies strictly between zj0 −1 and zj0 +1 . Let L± = dist(zj0 ±1 , I) > 0. Here if j0 = n, then L+ = ∞. Let r1 , . . . , rbj0 , . . . , rn , and s1 , . . . , sm be as in Lemma 3.7 except that we now require that rj0 ±1 < (|zj0 ±1 − zj0 ±2 | ∧ L± )/2. Let j0 −2 m−1 Y Y Q =L−2α − |zj − zj+1 | · |wm | −α (|wk | ∧ |wk − wk+1 |)−α j=1 k=1 Yn · (L+ ∧ |zj0 +1 − zj0 +2 |)−α (|zj − zj+1 | ∧ |zj − zj−1 |)−α . j=j0 +2 Here when m = 0, the |wm |−α (|wk | ∧ |wk − wk+1 |)−α disappears; and when j0 = n, the Qm−1 k=1 second line in the formula disappears. Let h ∈ (L+ ∧ L− )/2 and τhI = τI×[0,h] . Then Y Ym P[τhI < τrzjj < ∞, j ∈ Nn \ {j0 }; τhI < τswkk < ∞, k ∈ Nm ] ≲ Qh α rjα · sαk . (3.43) j∈Nm \{j0 } k=1 Proof. The proof is similar to that of Lemma 3.7. The only essential dierence is that now we do not get an upper bound of P[τhI < ∞] using (3.6). By symmetry we assume that zj 's 29 are positive and wk 's are negative. Let τ = τhI and zj0 = Re γ(τ ). Then zj0 is Fτ -measurable, and zj0 −1 < zj0 < zj0 +1 . Let E denote the event in (3.43). Then E = E∗τ ∩ E# , where E∗τ := {τ < τrzjj ∧ τz∗j , j ∈ Nn \ {j0 }; τ < τswkk ∧ τw∗ k , 1 ≤ k ≤ m} ∈ Fτ ; E# := {τrzjj < ∞, j ∈ Nn \ {j0 }; τswkk < ∞, 1 ≤ k ≤ m}. Suppose E∗τ occurs. Dene zej , Dj , D e j , rej , w ek , Ek , E ek , sek as in the previous proof. By DMP of chordal SLEκ and Proposition 3.1, we see that (3.31) also holds here. The estimates (3.37,3.38,3.40) still hold here by the same extremal length argument. Estimate (3.39) should be replaced by rej0 −1 rj0 −1 h rj −1 h 1∧ ≲ 2 ≤ 02 . (3.44) |e zj0 −1 | ∧ |e zj0 −1 − zej0 | |zj0 −1 − zj0 | L− When j0 = n, combining (3.37,3.38,3.40,3.44) with (3.31), we get α  s α jY 0 −2  α m−1  hr j0 −1 m rj Y sk α P[E# |Fτ , E∗τ ] ≲ · · L2− |wm | j=1 |zj − zj+1 | k=1 |wk − wk+1 | ∧ |wk | Taking expectation, we then get (3.43) in the case j0 = n. Suppose 2 ≤ j0 ≤ n − 1. Let Rj , j0 + 2 ≤ j ≤ n, be as in the proof of Lemma 3.7. We redene Rj0 +1 = (|zj0 +1 − zj0 +2 | ∧ L+ )/2. For each k = (kj0 +1 , . . . , kn ) ∈ (N ∪ {0})N(j0 ,n] , let Ek be dened as in the proof of Lemma 3.7 using the Rj , j0 + 1 ≤ j ≤ n dened here. By Remark 3.6, Y n P[Ek ] ≲ e−2αkj . (3.45) j=j0 +1 This inequality holds no matter whether all kj 's are zero or not. On the event Ek ∩ E∗ , the same extremal length argument shows that rej rj 1∧ ≲ , j0 + 1 ≤ j ≤ n. (3.46) |e zj − zej−1 | Rj e−kj 30 Combining (3.31,3.37,3.38,3.40,3.44) with (3.45,3.46) we get 0 −2  α jY n  hr j0 −1 rj α Y  r α j P[E ∩ Ek ] ≲ 2 · L− j=1 |zj − zj+1 | j=j +1 Rj 0  s α m−1 Y sk α Y n m · · · e−αkj |wm | k=1 |w k − w k+1 | ∧ |w k | j=j +1 0 Summing up the above inequality over k ∈ (N ∪ {0})N(j0 ,n] , we get (3.43) for j0 < n. Denition 3.9. Recall the Σn , n ∈ N, dened in Theorem 1.1. For z ∈ Σn and j0 ∈ Nn , we say that zj0 is an innermost component of z if there is no k ∈ Nn \ {j0 } such that zk lies strictly between 0 and zj0 . An element z ∈ Σn may have one or two innermost components. For z = (z1 , . . . , zn ) ∈ Σn , we dene the inner distance of z by d(z) := min{|zj − zk | : 0 ≤ j < k ≤ n}, where z0 := 0. Lemma 3.10. Let z ∗ = (z1∗ , . . . , zn∗ ) ∈ Σn . Suppose that z1∗ is an innermost component of z ∗ . Then for any ε > 0, there are δ ∈ (0, d(z ∗ )/3] and an H-hull H (depending on z ∗ and ε) such that ˆ {z ∈ H : |z − z1∗ | ≤ 3δ} ⊂ H ; ˆ dist(zj∗ , H) ≥ 3δ , 2 ≤ j ≤ n; and ˆ if z ∈ Σn and r ∈ (0, ∞)n satisfy ∥z − z ∗ ∥∞ ≤ δ and ∥r∥∞ ≤ δ , then Y n P[Kτrz1 ̸⊂ H; τrz11 < τrzjj < ∞, 2 ≤ j ≤ n] < ε rjα . (3.47) 1 j=1 Proof. Qn For r = (r1 , . . . , rn ) ∈ (0, ∞)n , let P (r) = j=1 rj . Fix a chordal SLEκ curve γ in H from 0 to ∞. For z = (z1 , . . . , zn ) ∈ Σn , r = (r1 , . . . , rn ) ∈ (0, ∞)n , and S ⊂ H, let z Er;S = {τrz11 < τrzjj < ∞, 2 ≤ j ≤ n; Kτrz1 ∩ S ̸= ∅}. 1 z Then (3.47) can be rewritten as P[Er;H\H ] < εP (r). 31 By Lemma 3.4, there is a positive continuous functions F∞ on Σn such that, for any z ∈ Σn and any r ∈ (0, ∞)n , z P[Er;{z∈H:|z|≥R} ] ≤ F∞ (z)R−α P (r), if ∥r∥∞ < d(z)/2 and R ≥ 2 max{|zk |}. (3.48) By Proposition 3.2, for any 2 ≤ k ≤ n, there are a constant β>0 and a positive continuous function Fk on Σn such that, for any z ∈ Σn , r ∈ (0, ∞)n , and r > 0, z P[Er;{z∈H:|z−z k |≤r} ] ≤ Fk (z)rβ P (r), if ∥r∥∞ < d(z)/8. (3.49) Note that, if ∥z − z ∗ ∥∞ ≤ d(z ∗ )/4, then d(z) ≥ d(z ∗ )/2 and max{|zj |} ≤ 2 max{|zj∗ |}. By (3.48,3.49) and the continuity of F∞ and Fk , 2 ≤ k ≤ n, there are R > 4 max{|zk∗ |} and r ∈ (0, d(z ∗ )/3) such that if ∥z − z ∗ ∥∞ ≤ d(z ∗ )/4, and ∥r∥∞ < d(z ∗ )/16, then z ε P[Er;{z∈H:|z|≥R}∪ Sn ] < P [r]. k=2 {z∈H:|z−zk |≤r} 2 We further assume that ∥z − z ∗ ∥∞ ≤ r/2. Then {z ∈ H : |z − zk∗ | ≤ r/2} ⊂ {z ∈ H : |z − zk | ≤ r} for 2 ≤ k ≤ n, which implies by the above formula that z ε P[Er;{z∈H:|z|≥R}∪ Sn ∗ ] < P [r], if ∥r∥∞ < d(z ∗ )/16. (3.50) k=2 {z∈H:|z−zk |≤r/2} 2 Since R > 2 max{|zk |} and r < d(z ∗ )/3, the semi-discs {z ∈ H : |z − zj∗ | ≤ r}, 1 ≤ j ≤ n, are mutually disjoint, and are all contained in the semi-disc {z ∈ H : |z| ≤ R}. By symmetry, we assume that z1∗ > 0. We relabel the components of z∗ by zj∗ , 1 ≤ j ≤ n′ , and wk∗ , 1 ≤ k ≤ m′ , where n′ ≥ 1, m′ ≥ 0, and n′ + m′ = n, such that wm ∗ ∗ ′ < · · · < w1 < 0 < z1∗ < · · · < zn∗ ′ . After relabeling, the symbol z1∗ still refers to the same point. Correspondingly, we relabel the components of every z ∈ Σn and r ∈ (0, ∞)n by zj , 1 ≤ j ≤ n′ , wk , 1 ≤ k ≤ m′ , rj , 1 ≤ j ≤ n′ , and sk , 1 ≤ k ≤ m′ . It is clear that, if ∥z − z ∗ ∥∞ < d(z ∗ )/2, then wm′ < · · · < w1 < 0 < z1 < · · · < zn′ , and so z1 is an innermost component of z. Dene compact intervals Ij , 2 ≤ j ≤ n, and Jk , 1 ≤ k ≤ m, as follows. If n′ = 1, we do not dene Ij 's. If n′ ≥ 2, let In′ = [zn∗ ′ + r/2, R], and Ij = [zj∗ + r/2, zj+1∗ − r/2], 32 2 ≤ j ≤ n′ − 1. If m = 0, we do not dene Jk 's. If m ≥ 1, let Jm′ = [−R, wm ∗ ′ − r/2], and ∗ Jk = [wk+1 + r/2, wk∗ − r/2], 1 ≤ k ≤ m′ − 1. If ∥z − z ∗ ∥∞ ≤ r/4, then the distance from every component of z to every interval Ij or Jk is at least r/4. By Lemma 3.8, there are continuous functions FIj , 2 ≤ j ≤ n′ , and F Jk , 1 ≤ k ≤ m′ , dened on the set of z ∈ Σn with ∥z −z ∗ ∥∞ ≤ r/4, such that, if ∥z −z ∗ ∥∞ ≤ r/4, ∥r∥∞ < r/8, and h < r/8, then for each 2 ≤ j ≤ n′ and 1 ≤ k ≤ m′ , z z P[Er;I j ×[0,h] ] < FIj (z)hα P (r), P[Er;J k ×[0,h] ] < FJk (z)hα P (r). Thus, there is h>0 such that, if ∥z − z ∗ ∥∞ ≤ r/4 and ∥r∥∞ ≤ r/8, then z z ε P[Er;I j ×[0,h] ], P[Er;J k ×[0,h] ]< P (r), 2 ≤ j ≤ n′ , 1 ≤ k ≤ m′ . (3.51) 2n Let n ′ m ′ [ [ H = {z ∈ H : |z| ≤ R} \ ({|z − zj∗ | ≤ r} ∪ Ij × [0, h]) \ ({|z − wk∗ | ≤ r} ∪ Jk × [0, h]). j=2 k=1 See Figure 3.5. Then H is an H-hull, which contains {z ∈ H : |z − z1∗ | ≤ r}, and the distance from each of z2∗ , . . . , zn∗ ′ and w1∗ , . . . , wm ∗ ′ to H is at least r. Combining (3.50) and (3.51), we z get P[Er;H\H ] < εP (r) if ∥z − z ∗ ∥∞ ≤ r/4, and ∥r∥∞ ≤ r/16. So we nd that (3.47) holds for such H and δ := r/16. 33 Figure 3.5: A gure for the proof of Lemma 3.10. This gure illustrates the ′ construction of the H-hull H in the proof of Lemma 3.10 in the case that n = 3 and m′ = 2. The H-hull H (the shaded region) is obtained by removing small discs of radius r centered at z2 , z3 , w1 , w2 and 4 rectangles with real interval bases and height h from the big semi-disc {z ∈ H : |z| ≤ R}. 34 CHAPTER 4 PROOF OF THE MAIN THEOREM We will nish the proof of Theorem 1.1 in this chapter. Recall that P denotes the law of a chordal SLEκ curve in H from 0 to ∞; and for z ∈ R \ {0} and r > 0, P∗z denotes the law of a two-sided chordal SLEκ curve in H from 0 to ∞ passing through z, and Prz denotes the conditional law P[·|τrz < ∞]. We will use an induction on n. By (1.7), Theorem 1.1 holds for n = 1. Let n ≥ 2. We make the induction hypothesis that Theorem 1.1 holds for n − 1. For any w = (w1 , . . . , wn−1 ) ∈ Σn−1 (Denition and s = (s1 , . . . , sn−1 ) ∈ (0, ∞)n−1 , we dene G(w, s) = P[τswj j < ∞, 1 ≤ j ≤ n]. (4.1) Qn−1 By the induction hypothesis, lims1 ,...,sn−1 →0+ j=1 s−α j G(w, s) = G(w). Given a chordal Loewner curve γ with the corresponding centered Loewner maps Zt 's, we dene a family of functions Gγt , t ≥ 0, on Σn−1 associated with γ by   Qn  |Zt′ (zj )|α G(Zt (z2 ), . . . , Zt (zn )), if t < τz∗j , 2 ≤ j ≤ n; j=2 Gγt (z2 , . . . , zn ) = (4.2)  0,  otherwise. When γ is a random Loewner curve, Gγt are random functions. We use E∗z1 [GTz1 (·)] to denote the expectation of Gγt (·) when γ follows the law P∗z1 , and t = Tz1 . Following the approach in [12], we will prove that for any 1 ≤ j0 ≤ n and z = (z1 , . . . , zn ) ∈ Σn , the following limit exists and is nite: n z Y Gj0 (z) := lim rj−α P[τrjj00 < τrzkk < ∞, ∀k ∈ Nn \ {j0 }]. (4.3) r1 ,...,rn →0+ j=1 It is clear that if the above limit exists and is nite for any 1 ≤ j0 ≤ n, then the same is true for the limit in (1.2), and we have X n G(z) = Gj (z). (4.4) j=1 In this chapter we will prove the following theorem. 35 Theorem 4.1. Given the induction hypothesis, for any 1 ≤ j0 ≤ n, the limit in (4.3) converges uniformly on any compact subset of Σn , and the limit function Gj0 is continuous on Σn . Moreover, we have Gj0 (z1 , . . . , zn ) = G(zj0 )E∗zj0 [GTzj (z1 , . . . , zbj0 , . . . , zn )], (4.5) 0 where the symbol zbj0 means that zj0 is omitted in the list from z1 to zn . It is clear that all statements of Theorem 1.1 in the induction step except for G ≍ F follow from Theorem 4.1 and (4.4). When we have the existence of G on Σn , the statement G≍F then follows immediately from Proposition 3.1 by sending r1 , . . . , rn to 0+ . After proving Theorem 4.1, we get a local martingale related to the Green's function. Corollary 4.2. For any xed z = (z1 , . . . , zn ) ∈ Σn , the process t 7→ Gγt (z) associated with a chordal SLEκ curve γ in H from 0 to ∞ is a local martingale up to τ := min{τz∗j , 1 ≤ j ≤ n}. Proof. Fix z = (z1 , . . . , zn ) ∈ Σn and let Mt = Gγt (z). It suces to prove that for any H-hull K , whose closure does not contain any of z1 , . . . , zn , M·∧TK is a martingale, where TK := inf{t > 0 : γ[0, t] ̸⊂ K}. The reason is that τ is the supremum of all such TK . To prove that M·∧TK is a martingale, we pick a small r > 0, and consider the martingale (r) Mt := r−nα P[τrzj < ∞, 1 ≤ j ≤ n|Ft ]. (r) By Theorem 4.1, DMP of chordal SLE and Koebe's distortion theorem, we have Mt → Mt on [0, τ ) as r → 0+ . We claim that the convergence is uniform on [0, TK ]. To see this, we apply Proposition 2.4 to conclude that there exist an H-hull H and a < b ∈ (0, ∞) such that (Zt (z1 ), . . . , Zt (zn )) ∈ H and a ≤ |Zt′ (zj )| ≤ b, 1 ≤ j ≤ n, for any t ∈ [0, TK ]. So we get the (r) uniform convergence of Mt → Mt over [0, TK ] by the uniform convergence of the n-point Green's function on the H-hull in H. So the claim is proved, which then implies that M·∧TK is a martingale, as desired. Remark 4.3. Qn We may write Mt = j=1 |gt′ (zj )|α G(gt (z1 ) − Ut , . . . , gt (zn ) − Ut ). If we know that G is C 2, then using Itô's formula and Loewner's equation (2.4), one can easily get the 36 following second order PDE for G: n n n κ  X 2 X 2 X −2 ∂zj G + ∂ zj · G + α 2 · G = 0. 2 j=1 j=1 z j j=1 zj Since the PDE does not depend on the order of points, it is also satised by the unordered Green's function G. We expect that the smoothness of G can be proved by Hörmander's theorem because the dierential operator in the above displayed formula satises Hörmander's condition. The rest part of the thesis is devoted to the proof of Theorem 4.1. By symmetry it suces to work on the case j0 = 1. We will rst prove in Section 4.1 the existence of G1 as well as the uniform convergence on compact subsets of Σn , and then prove in Section 4.2 the continuity of G1 . 4.1 Existence In this section, we work on the inductive step to prove the existence of the limit in (4.3) with j0 = 1. We now dene G1 on Σn using (4.5) instead of (4.3). In order to prove that the limit in (4.3) converges uniformly on each compact subset of Σn , it suces to show that, for any z ∗ = (z1∗ , . . . , zn∗ ) ∈ Σn and ε > 0, there exists δ>0 such that if z = (z1 , . . . , zn ) ∈ Σn and r = (r1 , . . . , rn ) ∈ (0, ∞)n satisfy that ∥z − z ∗ ∥∞ < δ and ∥r∥∞ < δ , then Yn | rj−α P[τrz11 < τrzjj < ∞, 2 ≤ j ≤ n] − G1 (z1 , . . . , zn )| < ε. (4.6) j=1 Fix z ∗ = (z1∗ , . . . , zn∗ ) ∈ Σn and ε > 0. Recall Denition 3.9. Let d∗ = d(z ∗ ). Let z = (z1 , . . . , zn ) ∈ Σn satisfy ∥z − z ∗ ∥∞ < d∗ /2. First suppose z1∗ is not an innermost component of z∗. Then z1 is not an innermost component of z. Then there is k0 ∈ {2, . . . , n} such that zk0 lies strictly between 0 and z1 . Under the law P∗z1 , we have τz∗k ≤ Tz1 , and so 0 GTz1 (z2 , . . . , zn ) = 0, which implies that G1 (z) = 0. On the other hand, by Lemma 3.7, Yn lim rj−α P[τrz11 < τrzkk < ∞, 2 ≤ k ≤ n] = 0, r1 ,...,rn →0+ j=1 37 and the convergence is uniform in some neighborhood of z∗. So we have (4.6) if z1∗ is not an innermost component of z∗. From now on, we assume that z1∗ is an innermost component of z∗. By symmetry we assume that z1∗ > 0. Let z = (z1 , . . . , zn ) ∈ Σn and r = (r1 , . . . , rn ) ∈ (0, ∞)n . Suppose ∥z − z ∗ ∥∞ < d∗ /4 and ∥r∥∞ < d∗ /4. Then the discs {|z − zj | ≤ rj }, 1 ≤ j ≤ n, are mutually disjoint. Let Er z denote the event {τrz11 < τrjj < ∞, 2 ≤ j ≤ n}. We will transform the rescaled probability Qn −α j=1 rj P[Er ] into G1 (z) (dened by (4.5)) in a number of steps. In each step we get an error term, and have an upper bound of the error term. We dene some good events depending on z. For any r > 0 and H-hull H , let Er;H denote the event that Kτrz1 ⊂ H . For R > r > s ≥ 0, let Er,s;R be the event that γ[τrz1 , τsz1 ] does not intersect the connected component of {z ∈ H : |z − z1 | = R} ∩ Hτrz1 which has z1 + R as an endpoint. e In the following, we use X≈Y to denote the approximation relation |X −Y | = e, and call z e the error term. Let z ′ = (z2 , . . . , zn ), r′ = (r2 , . . . , rn ), and Er′ = {τrjj < ∞, 2 ≤ j ≤ n}. For some H-hull H to be determined we use the following approximation relations: e∗1 P[Er ] ≈ P[Er ∩ Er1 ;H ] = P[τrz11 < ∞] · Erz11 [1Er1 ;H P[Er′ |Fτrz1 ]] 1 n e∗2 e∗3 Y ≈r1α G(z1 )Erz11 [1Er1 ;H P[Er′ |Fτrz1 ]] ≈ G(z1 )Erz11 [1Er1 ;H Gτrz1 (z ′ )] rkα . 1 1 k=1 We write G(r, ·) for Gτrz1 . For some η2 > η1 > r1 to be determined, we further use the following approximation relations: e4 G(z1 )Erz11 [1Er1 ;H G(r1 , z ′ )] ≈ G(z1 )Erz11 [1Er1 ;H ∩Eη1 ,r1 ;η2 G(r1 , z ′ )] e5 e6 ≈G(z1 )Erz11 [1Eη1 ;H ∩Eη1 ,r1 ;η2 G(r1 , z ′ )] ≈ G(z1 )Erz11 [1Eη1 ;H ∩Eη1 ,r1 ;η2 G(η1 , z ′ )] e7 e8 ≈G(z1 )Erz11 [1Eη1 ;H G(η1 , z ′ )] ≈ G(z1 )E∗z1 [1Eη1 ;H G(η1 , z ′ )] e9 e10 ≈G(z1 )E∗z1 [1Eη1 ;H ∩Eη1 ,0;η2 G(η1 , z ′ )] ≈ G(z1 )E∗z1 [1Eη1 ;H ∩Eη1 ,0;η2 G(0, z ′ )] e11 e12 ≈ G(z1 )E∗z1 [1E0;H ∩Eη1 ,0;η2 G(0, z ′ )] ≈ G(z1 )E∗z1 [1E0;H G(0, z ′ )] e13 ≈ G(z1 )E∗z1 [G(0, z ′ )] = G(z). 38 Qn Let ej = e∗j / k=1 rk , j = 1, 2, 3. Then Y n X 12 rk−α P[Er ] − G(z) ≤ ej . (4.7) k=1 j=1 Let τ = τrz11 , Dj = {z ∈ Hτ : |z − zj | ≤ rj }, zej = Zτ (zj ), D e j = Zτ (Dj ), re+ = radze (D ej ) j j and rej− = dist(e zj , ∂ D e j ∩ H), 2 ≤ j ≤ n. Let ze′ = (e z2 , . . . , zen ) and re′± = (er2± , . . . , ren± ). By Koebe distortion theorem, for any 2 ≤ j ≤ n, if rj < dist(zj , Kτ ), |Zτ′ (zj )|rj − + |Zτ′ (zj )|rj ≤ r j ≤ r j ≤ . (4.8) (1 + rj / dist(zj , Kτ ))2 (1 − rj / dist(zj , Kτ ))2 e e By DMP of chordal SLE and (4.1), G(e z ′ , re′− ) ≤ P[Er′ |Fτ , τ < τrzjj , 2 ≤ j ≤ n] ≤ G(e z ′ , re′+ ). (4.9) Let Cκ ∈ [1, ∞) be the constant in Proposition 2.10. By Lemma 3.10, there are a nonempty H-hull H and δH ∈ (0, d∗ /3], such that {z ∈ H : |z−z1∗ | ≤ 3δH } ⊂ H , dist(zj , H) ≥ 3δH , 2 ≤ j ≤ n, and whenever ∥z − z ∗ ∥∞ ≤ δH and ∥r∥∞ ≤ δH , we have n Y ε rj−α P[Erc1 ;H ∩ Er ] < . (4.10) j=1 11Cκ From now on, we always assume that ∥z−z ∗ ∥∞ < δH . Then H ⊃ {z ∈ H : |z−z1 | ≤ 2δH }, and dist(zj , H) ≥ 2δH , 2 ≤ j ≤ n. By (4.10), if ∥r∥∞ ≤ δH , ε e1 ≤ . 11Cκ Sending r2 , . . . , r n to 0+ in (4.10) and using Fatou's lemma, estimates (4.8,4.9) and the convergence of (n − 1)-point Green's function, we get ε r1−α P[τrz11 < ∞] · Erz11 [1Erc G(r1 , z ′ )] ≤ , if r1 ≤ δH . 1 ;H 11Cκ By Proposition 2.10, if r1 ≤ δH , ε r1−α P[τrz11 < ∞] · E∗z1 [1Erc G(r1 , z ′ )] ≤ . 1 ;H 11 39 Let r1 → 0+ . From r1−α P[τrz11 < ∞] → G(z1 ), Erc1 ;H → E0;H c , G(r1 , z) → G(0, z) and Fatou's lemma, we get G(z1 )E∗z1 [1E0;H c G(0, z ′ )] ≤ ε 11 , which implies ε e13 ≤ . 11 By Proposition 2.4, the set ΩH := {(gK (z2 ) − u, . . . , gK (zn ) − u) : K ∈ H(H), u ∈ SH , |zj − zj∗ | ≤ δH , 2 ≤ j ≤ n} is a compact subset of Σn−1 , and the set [n ′ QH := {|gK (z)| : K ∈ H(H), z ∈ [zj∗ − δH , zj∗ + δH ]} j=2 is a compact subset of (0, ∞). Let ξH = min{|wk | : w = (w2 , . . . , wn ) ∈ ΩH , 2 ≤ k ≤ n} > 0. For a ≥ 0, we write Z a (z ′ ) for (Zτaz1 (z2 ), . . . , Zτaz1 (zn )), when all components are well dened. We will use the fact that, on the event Ea;H , Z a (z ′ ) ∈ ΩH because Zτaz1 = gKτ z1 − Uτaz1 , a Kτaz1 ⊂ H , and Uτaz1 ∈ SKτ z1 ⊂ H by Corollary 2.6 and Proposition 2.2. Recall that we a assume that ∥z − z ∗ ∥∞ ≤ δH . So we have |Zτaz1 (zj )| ≥ ξH , 2 ≤ j ≤ n, on the event Ea;H . (4.11) By the continuity of (n − 1)-point Green's function and the compactness of ΩH and QH , we see that, for any a ≥ 0, G(a, z ′ ) is bounded by a constant depending only on κ, n, z ∗ , H, δH on the event Ea;H . By (4.8,4.9), the compactness of ΩH and QH , and Proposition 3.1, P[Er′ |Fτrz1 , Er1 ;H ] is 1 Qn bounded by α j=2 rj times some constant depending only on κ, n, z ∗ , H, δH . By (1.7) and the above bound, there are β1 ∈ (0, ∞) depending only on κ and C1H ∈ (0, ∞) depending only on κ, n, z ∗ , H, δH such that, if r1 < |z1 |, e2 ≤ C1H r1β1 . Since the convergence of (n − 1)-point ordered Green's function is uniform over compact ′ sets, by (4.8,4.9) and the compactness of ΩH and QH , we nd that there is δH ∈ (0, δH ) 40 ′ depending only on κ, n, H, δH such that if ∥r∥∞ ≤ δH , then ε e3 < . 11 Since G is continuous on Σn−1 , by the compactness of ΩH and QH , G(a, z ′ ) is bounded by some constant depending only on κ, n, z ∗ , H, δH on the event Ea;H . Combining this fact for a ∈ {r1 , η1 , 0} with Proposition 3.3 and the boundedness of G(z1 ) (over [z1∗ − δH , z1∗ + δH ]), we nd that there is C2H ∈ (0, ∞) depending only on κ, n, z ∗ , H, δH such that e4 , e7 , e9 , e12 ≤ C2H (η1 /η2 )α . Since H ⊃ {z ∈ H : |z − z1∗ | ≥ 3δH }, if η2 ≤ 2δH , then Er1 ;H ∩ Eη1 ,r1 ;η2 = Eη1 ;H ∩ Eη1 ,r1 ;η2 and Eη1 ;H ∩ Eη1 ,0;η2 = E0;H ∩ Eη1 ,0;η2 , which implies that e5 = e11 = 0. Combining Proposition 2.11 with the boundedness of G(z1 ) and G(η1 , z ′ ) on the event Eη1 ;H , we nd that there are β2 > 0 depending only on κ and C3H ∈ (0, ∞) depending only on κ, n, z ∗ , H, δH such that, if r1 < η1 /6, e8 ≤ C3H (r1 /η1 )β2 . Recall that Yn Y n ′ G(η1 , z ) = |Zτ′ ηz1 (zj )|α · G(Z η1 (z )), ′ ′ G(r1 , z ) = |Zτ′ rz1 (zj )|α · G(Z r1 (z ′ )). 1 1 j=2 j=2 Assume η2 ≤ 2δh . Then Er1 ;H ∩ Eη1 ,r1 ;η2 = Eη1 ;H ∩ Eη1 ,r1 ;η2 , and on this common event, Z η1 (z ′ ), Z r1 (z ′ ) ∈ ΩH . Let K∆ = Kτrz1 /Kτηz1 . On the event Eη1 ,r1 ;η2 , by Proposition 2.9, 1 1 diam(K∆ ) ≤ 8η2 , and by Proposition 2.7, we have ∥Z η1 (z ′ ) − Z r1 (z ′ )∥∞ ≤ 56η2 . (4.12) From K∆ = Kτrz1 /Kτηz1 we know gτrz1 = gK∆ ◦ gτηz1 . Let Z∆ = gK∆ (· + Uτηz1 ) − Uτrz1 . Then 1 1 1 1 1 1 ′ ′ Zτrz1 = Z∆ ◦ Zτηz1 and Z∆ (z) = gK ∆ (· + Uτηz1 ). By Proposition 2.5, Uτηz1 ∈ K∆ . By Proposition 1 1 1 1 2.3, for z ∈ H,  8η 2 ′ 2 |Z∆ (z) − 1| ≤ 5 , if |z| ≥ 40η2 . (4.13) |z| 41 Let zej = Zτηz1 (zj ), 2 ≤ j ≤ n. Then Zτ′ rz1 (zj ) = Z∆ ′ zj ) · Zτ′ ηz1 (zj ), (e and by (4.11) |e zj | ≥ 1 1 1 ′ 2 ξH on the event Eη1 ;H . Thus, if η2 ≤ ξH /40, then |Z∆ (zj ) − 1| ≤ 320η22 /ξH . Since G is continuous on Σn−1 , it is uniformly continuous on the compact set ΩH . By (4.12), |G(Z η1 (z ′ )) − G(Z r1 (z ′ )| → 0 uniformly as η2 → 0+ . Combining these facts with the com- ′′ ′ pactness of QH and the expressions of G(η1 , ·) and G(r1 , ·), we nd that, there is δH ∈ (0, δH ) depending only on κ, n, z ∗ , H, δH such that, if η 2 ≤ δH ′′ , then ε e6 , e10 < . 11 We now explain how to choose the H and η1 , η2 in the approximation with errors from e1 to e13 . First, we choose the H-hull H and δH > 0 such that e1 , e13 ≤ ε 11 if ∥z − z ∗ ∥∞ ≤ δH ′ ′′ and ∥r∥∞ ≤ δH . We have the quantities C1H , C2H , C3H , δH , δH ∈ (0, ∞) depending only on ′ ′′ κ, n, z ∗ , H, δH . Assume that ∥z − z ∗ ∥∞ ≤ δH . If ∥r∥∞ ≤ δH , then e3 < ε 11 . Let η2 = δH . ε ′′ Then we have e6 , e10 < 11 . Since δH < δH , we have η2 < δH , and so e5 = e11 = 0. Let ε ε η1 = (ε/(11C2H ))1/α η2 . Then e4 , e7 , e9 , e12 ≤ 11 . If r1 < (ε/(11C1H ))1/β1 , then e2 < 11 ; and if r1 < (ε/(11C3H ))1/β2 η1 , then e8 < ε 11 . In conclusion, if ∥z − z ∗ ∥∞ ≤ δH and ′  ε 1/β2  ε 1/α ′′  ∥r∥∞ < δH ∧ · · δ H =: δ, 11C3H 11C2H then e5 = e11 = 0 and all ej 's are bounded by ε/11, which then imply by (4.7) that (4.6) holds. Thus, we get the existence of the limit in (4.3) with j0 = 1 as well as the uniform convergence on compact subsets of Σn . 4.2 Continuity In this section, we prove the continuity of the function G1 on Σn . We adopt the notation in the previous section. By the rescaling property and left-right symmetry of SLE, for any c ∈ R \ {0}, z = (z1 , . . . , zn ) ∈ Σn , and r > 0, cz1 czk P[τrz1 < τrzk < ∞, 2 ≤ k ≤ n] = P[τ|c|r < τ|c|r < ∞, 2 ≤ k ≤ n]. Multiplying both sides by r−nα and sending r to 0+ , we get by the existence of the limit in (4.3) that G1 (z) = |c|nα G1 (cz). In particular, we have G1 (z) = |z1 |−nα G1 (1, z2 /z1 , . . . , zn /z1 ). 42 Thus, it suces to prove that G1 (1, ·) is continuous on Σ1n−1 , which is the set of w ∈ Σn−1 such that (1, w) ∈ Σn . Dene G b on Σ1n−1 such that G(w) b = E∗1 [GT1 (w)]. Then G1 (1, w) = G(1)G(w) b . So it suces to prove that Gb is continuous on Σ1n−1 . From the previous section, Gb vanishes on the set of w which has at least one component lying in (0, 1). Since such set is open in Σ1n−1 , it suces to prove the continuity of G b at other points of Σ1n−1 . Fix w∗ = (w2∗ , . . . , wn∗ ) ∈ Σ1n−1 such that wk∗ ̸∈ (0, 1) for any 2 ≤ k ≤ n. Let w0 = 0 and w1 = 1 . Let d∗ = min{|wj∗ − wk∗ | : 0 ≤ j < k ≤ n} > 0. Let ε > 0. By the argument of an upper bound of e13 in the previous section, there are δH ∈ (0, d∗ /3) and an H-hull H, such that {z ∈ H : |z − 1| ≤ 3δH } ⊂ H , dist(wj∗ , H) ≥ 3δH , 2 ≤ j ≤ n, and for any w = (w2 , . . . , wn ) ∈ Rn−1 satisfying ∥w − w∗ ∥∞ ≤ δH , we have E∗1 [1E0;H c GT1 (w)] < ε/3, where E0;H is the event that γ[0, T1 ] ⊂ H . Suppose ∥w − w∗ ∥∞ ≤ δH , we use the following approximation relations for such H: e1 e2 e3 G(w) b = E∗1 [GT1 (w)] ≈ E∗1 [1EH GT1 (w)] ≈ E∗1 [1EH GT1 (w∗ )] ≈ E∗1 [GT1 (w∗ )] = G(w b ∗ ). We have known that e1 , e3 < ε/3. It remains to bound e2 . Qn We write Z(w) = (ZT1 (w2 ), . . . , ZT1 (wn )). Then GT1 (w) = j=2 ZT′ 1 (wj )α G(Z(w)). As w → w∗ , we have G(Z(w)) → G(Z(w∗ )) by the continuity of (n − 1)-point Green's function, and ZT′ 1 (wj ) → ZT′ 1 (wj∗ ), 2 ≤ j ≤ n, which together imply that GT1 (w) → GT1 (w∗ ). We now show that the convergence is uniform (independent of the randomness) on the event E0;H . By the previous section, on the event E0;H , we have Z(w), Z(w∗ ) ∈ ΩH , and ZT′ 1 (wj ), ZT′ 1 (wj∗ ) ∈ QH , 2 ≤ j ≤ n. By the compactness of QH , on the event E0;H , the random map ZT1 is equicontinuous (independent of the randomness) on [wj − δH , wj + δH ] for 2 ≤ j ≤ n. Thus, as w → w∗ , Z(w) → Z(w∗ ) uniformly on the event E0;H . Since G is uniformly continuous on the compact set ΩH , we get G(Z(w)) → G(Z(w∗ )) uniformly on the event E0;H as w → w∗ . By Koebe's distortion theorem, for 2 ≤ j ≤ n, ZT′ 1 (wj ) → ZT′ 1 (wj∗ ) uniformly on the event E0;H as w → w∗ . Thus, GT1 (w) → GT1 (w∗ ) uniformly on the event E0;H as w → w∗ . In ′ particular, there is δH ∈ (0, δH ) such that if ∥w − w∗ ∥∞ ≤ δH ′ ∗ , then |GT1 (w) − GT1 (w )| < ε/3 43 on the event E0;H , which implies that e2 < ε/3. Thus, if ∥w − w∗ ∥∞ ≤ δH ′ , then b ∗ )| ≤ e1 + e2 + e3 < ε ε ε |G(w) b − G(w + + = ε. 3 3 3 So we get the desired continuity of G b at w∗ . 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