@article {29412,
	title = {Quantitative Russo-Seymour-Welsh for random walk on random graphs and decorrelation of UST},
	journal = {Journal of Theoretical Probability},
	year = {Accepted},
	abstract = {<blockquote>
	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.
</blockquote>
},
	url = {https://arxiv.org/abs/2106.10323},
	author = {Gourab Ray and Tingzhou Yu}
}
@mastersthesis {29442,
	title = {The coupling of uniform spanning trees and quantitative Russo-Seymour-Welsh for random walk on random graphs},
	year = {2021},
	type = {M.Sc. thesis},
	abstract = {The central concern of this thesis is the study of the Russo-Seymour-Welsh (RSW) theory. The first contribution of this thesis is a macroscopic decorrelation result for uniform spanning trees (USTs) on random planar graphs based on the RSW assumption. A similar result was established on a fixed graph in [BLR20, Theorem 4.21]. We extend this result to USTs on random graphs. In particular, we show that a similar coupling can be obtained for a collection of graphs, which has a high probability. This is the key missing step in the application of the proof strategy in [BLR20] for random graphs, which established the scaling limits of height function of dimer model to a Gaussian free field on a fairly general class of fixed graphs. The second contribution of this thesis is the RSW type results for random walks on two concrete and natural examples: the unique infinite cluster of supercritical bond percolation in $\Z^2$ and the Poisson-Delaunay triangulation in $\R^2$. We show that random walks crossing a rectangle without exiting occurs with a stretched exponentially high in the scale. The main tool used in the proof is heat kernel estimates for random walks on the supercritical bond percolation. The proof of RSW for bond percolation is a quick application of a combination of Barlow{\textquoteright}s results. However, we cannot apply Barlow{\textquoteright}s results for the Delaunay triangulation directly since there is no uniform bound on degree. A key input is a quantitative isoperimetric inequality for the Delaunay triangulation, which we consider to be another novel contribution of this thesis.},
	url = {http://hdl.handle.net/1828/13316},
	author = {Tingzhou Yu}
}
