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Boundary Correlations in Planar LERW and UST Boundary correlations in planar LERW and UST Alex Karrila [email protected] Department of Mathematics and Systems Analysis P.O. Box 11100, FI-00076 Aalto University, Finland Kalle Kytölä [email protected] Department of Mathematics and Systems Analysis P.O. Box 11100, FI-00076 Aalto University, Finland https://math.aalto.fi/~kkytola/ Eveliina Peltola [email protected] Section de Mathématiques, Université de Genève 2–4 rue du Lièvre, Case Postale 64, 1211 Genève 4, Switzerland Abstract. We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple SLEκ at κ = 2. Contents 1. Introduction 2 1.1. Boundary visit probabilities of the loop-erased random walk 3 1.2. Connectivity probabilities of boundary branches in the uniform spanning tree 5 arXiv:1702.03261v3 [math-ph] 17 Sep 2019 1.3. Steps of the proof(s) 7 1.4. Multiple SLE2 8 1.5. Beyond the present work: conformal blocks and q-deformations 8 1.6. Organization of the article 9 2. Combinatorics of link patterns and the parenthesis reversal relation 9 2.1. Combinatorial objects and bijections 10 2.2. Partial order and the parenthesis reversal relation 11 2.3. Dyck tilings and inversion of weighted incidence matrices 14 2.4. Wedges, slopes, and link removals 19 1 2 2.5. Cascades of weighted incidence matrices 24 2.6. Inverse Fomin type sums 26 3. Uniform spanning trees and loop-erased random walks 30 3.1. Graphs embedded in a planar domain 30 3.2. The planar uniform spanning tree 31 3.3. Relation between uniform spanning trees and loop-erased random walks 34 3.4. Fomin’s formula 37 3.5. Solution of the connectivity partition functions 39 3.6. Random spanning trees on weighted graphs 41 3.7. Scaling limits 41 3.8. Generalizations of the main results 44 4. Relation to multiple SLEs 47 4.1. Multiple SLEs 48 4.2. Proof of Theorem 4.1 51 5. Boundary visit probabilities and third order PDEs 54 5.1. Scaling limit of boundary visit probabilities 55 5.2. Third order partial differential equations through fusion 59 5.3. Proof of Theorem 3.17 65 5.4. Asymptotics of the scaling limits of the boundary visit probabilities 65 Appendix A. Example boundary visit formulas 69 A.1. One boundary visit 69 A.2. Two boundary visits 69 References 70 1. Introduction In this article, we consider the planar loop-erased random walk (LERW) and a closely related model, the planar uniform spanning tree (UST). We find explicit expressions for probabilities of certain connectivity and boundary visit events in these models, and prove that, in the scaling limit as the lattice spacing tends to zero, these observables converge to conformally covariant functions, which satisfy systems of second and third order partial differential equations (PDEs) predicted by conformal field theory (CFT) [BPZ84b, BSA88, BB03, BBK05, KP19, Dub15a, Dub15b, JJK16]. As a byproduct, we also obtain formulas for the pure partition functions of multiple Schramm-Loewner evolutions (SLEs), and the existence of all extremal local multiple SLEs at the specific parameter value κ = 2 corresponding to these models [BBK05, Dub07, KP16]. We now give a brief overview of the main results of this article. The detailed formulations of the statements made here are given in the bulk of the article, as referred to below. 3 Figure 1.1. A loop-erased random walk on the square grid and the underlying random walk (in grey). 1.1. Boundary visit probabilities of the loop-erased random walk. Throughout, we let G = (V; E) be a finite connected graph with a distinguished non-empty subset @V ⊂ V of vertices, called boundary vertices. We assume that G is a planar graph embedded in a Jordan domain Λ ⊂ C such that the boundary vertices @V lie on the Jordan curve @Λ. We denote by @E ⊂ E the set of edges e = he@ ; e◦i which connect a boundary vertex e@ 2 @V to an interior vertex e◦ 2 V◦ := V n @V. Our scaling limit results are valid in any setup where the random walk excursion kernels and their discrete derivatives on the graphs converge to the Brownian excursion kernels and their derivatives. To be specific, we present the results in the following square grid approximations (see Figure 3.7). We fix a Jordan domain Λ and a number of boundary points p1; p2;::: and p^1; p^2;::: on horizontal or vertical parts of the boundary @Λ. For any small δ > 0, we let Gδ = (Vδ; Eδ) be a subgraph of the square lattice δZ2 of mesh size δ naturally approximating the domain Λ (as detailed in Section 3). We also let δ δ δ δ ej 2 @E be the boundary edge nearest to pj, and let e^s 2 E be the edge at unit graph distance from the boundary nearest to p^s. By scaling limits we mean limits of our quantities of interest as δ ! 0 in this setting. Generally, a loop-erased random walk (LERW) is a simple path on the graph obtained by erasing loops from a random walk in their order of appearance — see Figure 1.1 for illustration and Section 3 for the @ ◦ @ ◦ precise definitions. Choose two boundary edges ein = hein; eini and eout = heout; eouti. By the LERW ◦ λ on G from ein to eout we mean the loop-erasure of a random walk started from the vertex ein and conditioned to reach the boundary via the edge eout. In the present article, we find an explicit formula for the probability of the event that the LERW λ passes through given edges e^1;:::; e^N 0 at unit distance from the boundary, as illustrated in Figure 1.2. Moreover, we prove that this probability converges in the scaling limit to a conformally covariant function ζ, which satisfies a system of second and third order partial differential equations, as has been previously predicted based on conformal field theory. δ δ δ δ δ δ δ Theorem (Theorem 3.17). Let G with ein; eout 2 @E and e^1;:::; e^N 0 2 E be a square grid approxima- tion of a domain Λ with marked boundary points pin; pout; p^1;:::; p^N 0 . The probability that the LERW δ δ δ δ δ δ λ on G from ein to eout passes through the edges e^1;:::; e^N 0 has the following conformally covariant scaling limit: N 0 1 δ δ δ Y 0 3 ζ φ(pin); φ(^p1); : : : ; φ(^pN 0 ); φ(pout) 0 Peδ ;eδ λ uses e^1;:::; e^N 0 −! jφ (^ps)j × ; δ3N in out δ!0 −2 s=1 φ(pout) − φ(pin) 4 ein e^1 e^2 eout e^3 e^4 Figure 1.2. Illustration of a loop-erased random walk passing through edges at unit distance from the boundary. We study the probabilities of such boundary visit events. where φ:Λ ! H is a conformal map from the domain Λ to the upper half-plane H = fz 2 C j =m(z) > 0g. The function ζ(xin;x ^1;:::; x^N 0 ; xout) is a positive function of the real points xin; x^1; ··· ; x^N 0 ; xout, sat- isfying the following system of linear partial differential equations: the N 0 third order PDEs " @3 8 @2 8 @ 12 @ 24 3 + − 2 − 2 + 3 @x^s xin − x^s @xin @x^s (xin − x^s) @x^s (xin − x^s) @xin (xin − x^s) 8 @2 8 @ 12 @ 24 + − 2 − 2 + 3 xout − x^s @xout @x^s (xout − x^s) @x^s (xout − x^s) @xout (xout − x^s) # X Å 8 @2 24 @ 12 @ 72 ã + − 2 − 2 + 3 ζ = 0 x^t − x^s @x^t@x^s (^xt − x^s) @x^s (^xt − x^s) @x^t (^xt − x^s) 1≤t≤N 0 t6=s for s = 1;:::;N 0, as well as the second order PDE " N 0 N 0 # @2 2 @ 2 X 2 @ X 6 + − + − ζ = 0 @x2 x − x @x (x − x )2 x^ − x @x^ (^x − x )2 in out in out out in t=1 t in t t=1 t in and another second order PDE obtained by interchanging the roles of xin and xout above. We make some remarks concerning the above result. The scaling limit of the LERW path λδ above is a conformally invariant random curve called the chordal SLEκ with parameter κ = 2, by some of the earliest of the celebrated works showing convergence of lattice model interfaces to SLEs [LSW04, Zha08], see also [CS11, YY11]. The scaling limit of the probability that a LERW uses one particular edge in the interior of the domain has been studied in [Ken00, Law14], see also [KW15]. The most accurate results currently known [BLV16] state that this probability divided by δ3=4 converges in the scaling limit to a conformally covariant expression known as the SLE Green’s function at κ = 2. This has been recently used to prove the convergence of the LERW to the chordal SLEκ with κ = 2 in a sense that includes parametrization [LV16]. In contrast to the probability of visiting an interior edge, our scaling limit results concern probabilities of visits to an arbitrary number of edges at unit distance from the boundary.
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