Quantitative Russo-Seymour-Welsh for random walk on random graphs and decorrelation of UST

TL;DR

This paper establishes a quantitative Russo-Seymour-Welsh (RSW) estimate for random walks on certain random planar graphs, enabling new results on decorrelation of uniform spanning trees and Gaussian free field limits.

Contribution

It provides the first quantitative RSW-type result for random walks on random planar graphs, crucial for analyzing uniform spanning trees and related models.

Findings

Proves stretched exponential bounds for crossing probabilities of random walks.

Derives near optimal decorrelation results for uniform spanning trees.

Discusses implications for Gaussian free field scaling limits.

Abstract

We prove a quantitative Russo-Seymour-Welsh (RSW) type result for random walks on two natural examples of random planar graphs: the supercritical percolation cluster in the square lattice and the Poisson Voronoi triangulation in the plane. More precisely, we prove that the probability that a simple random walk crosses a rectangle in the hard direction with uniformly positive probability is stretched exponentially likely in the size of the rectangle. As an application we prove a near optimal decorrelation result for uniform spanning trees for such graphs. This is the key missing step in this setup while applying of the proof stretegy of a previous article on universality of dimers ("Dimers and imaginary geometry." Ann. Probab. 48 (1) 1 - 52) where random walk RSW was assumed to hold with probability 1. Applications to almost sure Gaussian free field scaling limit for dimers on Temperleyan type modification on such graphs are also discussed.