High-dimensional critical percolation and critical Branching random walks are intersection equivalent

Bruno Schapira

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arXiv · 2610.37133

High-dimensional critical percolation and critical Branching random walks are intersection equivalent

Bruno Schapira

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Abstract

We prove that on $\mathbb Z^d$, with $d>6$, and under the usual assumption that the two-point function and the Green's function are of the same order, the probabilities to hit any finite set for a critical Branching random walk starting from the origin and the critical percolation cluster of the origin are within a positive multiplicative constant one from each other, as long as the set is sufficiently far away from the origin comparatively to its diameter. The proof relies on the introduction of a truncated equilibrium measure in percolation and the derivation of an approximate last passage formula.

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Bruno Schapira. 2026-10-27. High-dimensional critical percolation and critical Branching random walks are intersection equivalent. https://arxiv.org/abs/2610.37133

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