Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve

Abstract

In this paper we consider the 'natural' random walk on a planar graph and scale it by a small positive number δ. Given a simply connected domain D and its two boundary points a and b, we start the scaled walk at a vertex of the graph nearby a and condition it on its exiting D through a vertex nearby b, and prove that the loop erasure of the conditioned walk converges, as δ ↓ 0, to the chordal SLE 2 that connects a and b in D, provided that an invariance principle is valid for both the random walk and the dual walk of it. Our result is an extension of one due to Dapeng Zhan [12] where the problem is considered on the square lattice. A convergence to the radial SLE 2 has been obtained by Lawler, Schramm and Werner [3] for the square and triangular lattices and by Yadin and Yehudayoff [10] for a wide class of planar graphs. Our proof, though an adaptation of that of [3] and [10], involves some new ingredients that arise from two sources: one for dealing with a martingale observable that is different from that used in [3] and [10] and the other for estimating the harmonic measures of the random walk started at a boundary point of a domain. MSC2010 subject classifications. 60F17, 60J67, 82B41.

Boundary behaviour of RW's on planar graphs and convergence of LERW to chordal SLE$_2$

“…The primary contribution of the present paper is such an estimate (Proposition 4.6) that states that the excursion leaves an intrinsic neighborhood of its initial point not 'along' the boundary but through an intrinsic interior of the domain with high probability. The result plays a key role in our verification of the convergence of LERW to the chordal SLE: it refines the result of [12] by removing the second extra assumption mentioned above.…”

“…Suzuki [12] proves the theorem above under the additional assumption that ∂D is locally analytic at v 0 and S x satisfy invariance principle uniformly for the initial point x ∈ V . As is remarked in [12] the first assumption of the local analiticity at v 0 is imposed to assure the conclusion of Proposition 4.6 , while the uniformity of invariance principle with respect to initial points of the random walk may be replaced by our hypothesis (H) owing to the results of [16] that are cited as Propositions 2.1 and 4.10 in the present paper.…”

“…In what follows we present main steps of the proof of Theorem 5.5, which is similar to those found in [14] or [12], focusing the description on the new ingredients that are special to the present setting.…”

“…Yadin and Yehudayoff [16] extend the result of [6], the convergence of LERW to a radial SLE to that for the natural random walks on planar graphs under a natural setting (the same as ours) of the problem. Recently Suzuki [12] have obtained a chordal version of their result: the LERW in a simply connected domain conditioned to connect two boundary vertices converges to a chordal SLE 2 curve in a setting similar to [16] under the assumptions (1) the invariance principle holds uniformly for starting points of the walk and more seriously (2) the boundary of the domain is locally analytic at the starting boundary point of the random walk from whose trajectory (or rather its time reversal) the LERW is derived.…”

“…In this paper we are concerned with the chordal case of LERW on a planar graph as in [12]. For the chordal case of LERW we need unlike the radial case to deal with an excursion of random walk (a random walk path in a domain connecting a boundary point with another one), and we are forced to estimate the harmonic measure of the random walk started at a vertex on (or near) the boundary and conditioned to exit the domain through another boundary vertex that is specified in advance.…”