TY - JOUR
AU1 - Ray, Gourab
AU2 - Yu, Tingzhou
AB - 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 Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {Z}^2$$\end{document} and the Poisson Voronoi triangulation in R2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^2$$\end{document}. 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 the application of the proof strategy of Berestycki et al. (Ann Probab 48(1):1–52, 2020) for such graphs [in Berestycki et al. (2020), 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.
TI - Quantitative Russo–Seymour–Welsh for Random Walk on Random Graphs and Decorrelation of Uniform Spanning Trees
JF - Journal of Theoretical Probability
DO - 10.1007/s10959-023-01248-7
DA - 2023-12-01
UR - https://www.deepdyve.com/lp/springer-journals/quantitative-russo-seymour-welsh-for-random-walk-on-random-graphs-and-ne9so2gS8S
SP - 2284
EP - 2310
VL - 36
IS - 4
DP - DeepDyve
ER -