Solidification Estimates for Random Walks on Supercritical Percolation Clusters

Abstract We consider the simple random walk on the infinite cluster of a general class of percolation models on $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , $$d\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain “porous interfaces” surrounding the discrete blow-up of a compact set A . These controls substantially generalize previous results obtained in Nitzschner and Sznitman (J. Eur.Math. Soc. (JEMS) 22 (8), 2629–2672, 2020) for Brownian motion in $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and in Chiarini and Nitzschner (Comm. Math. Phys. 386 (3), 1685–1745, 2021) for random walks on $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.