We consider the simple random walk on the infinite cluster of a general class of percolation models on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, 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, <CitationRef CitationID="CR36">2020</CitationRef>) for Brownian motion in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and in Chiarini and Nitzschner (Comm. Math. Phys. 386(3), 1685–1745, <CitationRef CitationID="CR16">2021</CitationRef>) for random walks on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

Solidification Estimates for Random Walks on Supercritical Percolation Clusters

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2026年4月21日

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Alberto Chiarini,
Zhizhou Liu,
Maximilian Nitzschner

摘要

We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb {Z}^d$ $Z^{d}$ , $d\ge 3$ $d \geq 3$ , 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 $\mathbb {R}^d$ $R^{d}$ and in Chiarini and Nitzschner (Comm. Math. Phys. 386(3), 1685–1745, 2021) for random walks on $\mathbb {Z}^d$ $Z^{d}$ equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.