Quantitative Russo–Seymour–Welsh for Random Walk on Random Graphs and Decorrelation of Uniform Spanning Trees

• Published in: Journal of theoretical probability Vol. 36; no. 4; pp. 2284 - 2310
• Main Authors: Ray, Gourab, Yu, Tingzhou
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Graph theory; Graphs; Mathematics; Mathematics and Statistics; Percolation; Probability Theory and Stochastic Processes; Random walk; Statistics; Trees (mathematics); Triangulation; Gaussian-free field; Random environment; Dimer; Uniform spanning tree
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-023-01248-7

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 Z 2 and the Poisson Voronoi triangulation in R 2 . 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.