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arXiv · 2610.35553

On the number of pioneers in high-dimensional critical percolation

Amine Asselah·Bruno Schapira
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Abstract

We prove that for critical percolation in high dimension the expected number of pioneers on any set containing the origin is of the same order as the probability that a simple random walk starting from the origin exits the set. Our proof works in all dimensions $d>6$, under the usual assumption that the two-point function is of the same order as the Green's function. As an application we obtain a sharp uniform quantitative bound on the probability that the cluster of the origin intersects any given ball.

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Amine Asselah, Bruno Schapira. 2026-10-26. On the number of pioneers in high-dimensional critical percolation. https://arxiv.org/abs/2610.35553

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