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

Abstract

This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected grid domain and conditioned to exit it through another boundary vertex specified in advance. The primary contribution of the present paper is an estimate, which states that the excursion of the conditioned walk leaves an intrinsic neighborhood of its initial point not 'along' the boundary but through an intrinsic interior of the domain with high probability. Based on this result we give a proof for the convergence to the chordal SLE, a result that has recently been proved by Suzuki [12] under an analyticity assumption on the boundary of the domain arising in the limit.

The dimer model on Riemann surfaces, I

“…Using Wilson's algorithm we patch up several pieces in which we can pretend the walk lives in a simply connected neighbourhood. For this we take full advantage of a relatively recent chordal version of the convergence of LERW to SLE 2 due to Uchiyama [41].…”

“…However, locally, such a loop-erased random walk will behave as if on a portion of the plane where the scaling limit is known. Indeed in this situation, the assumptions on Γ #δ in Section 5.1 and a result of Yadin and Yehudayoff [42] as well as Uchiyama [41] guarantee the convergence of a small portion of the path towards an SLE 2 -type curve (we need Uchiyama's result to deal with rough boundaries induced by the past of loop-erased random walk itself). It simply remains to glue these pieces together.…”

“…Theorem 5.13 (Uchiyama [41]). Let D be a proper simply connected domain such that D ⊂ D. Let D#δ be a sequence of simply connected discrete domains with D #δ being its interior.…”

“…This follows from the work of Uchiyama. We emphasise here that Uchiyama uses a locally uniform invariance principle (called hypothesis (H) in [41], Section 2), which says that for any compact set, a random walk run up to the time it leaves the compact set is uniformly close to a Brownian motion (up to reparametrised curves) and the uniformity is over any starting point in the compact set. This is clearly satisfied in our case, see Remark 5.1.…”

On the convergence of massive loop-erased random walks to massive SLE(2) curves

“…It is worth noting that in [34] it was assumed that the boundary of Ω is 'flat' near the target point b, a technical restriction which was removed in [29] in the general setup of [33]. Our approach to this technicality is based upon the tools from [6] (see Section 3.2 for details), similar uniform estimates were independently obtained by Karrila [13, Appendix A] basing upon the conformal crossing estimates developed for the random walk in [16].…”