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

Gourab Ray () and Tingzhou Yu ()
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Gourab Ray: University of Victoria
Tingzhou Yu: University of Victoria

Journal of Theoretical Probability, 2023, vol. 36, issue 4, 2284-2310

Abstract: 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 $$\mathbb {Z}^2$$ Z 2 and the Poisson Voronoi triangulation in $$\mathbb {R}^2$$ 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.

Keywords: Dimer; Random environment; Uniform spanning tree; Gaussian-free field (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-023-01248-7

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