  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.
