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arXiv · 2610.08682

Non-perturbative finite connectivities in supercritical Bernoulli percolation on $\mathbb{Z}^d$ with $d\geq 3$

Yacine Aoun·Kamil Khettabi
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Abstract

We solve a longstanding problem in Bernoulli percolation: we derive sharp Ornstein--Zernike asymptotics for the probability that two distant points are connected by a finite cluster in the whole supercritical regime. Our proof has two main ingredients. First, the renewal structure from Fridbergh and Hammond (2024) yields an a priori bound on cluster volume. Second, we adapt the coarse-graining and irreducible-cluster framework of Companion, Ioffe, Velenik (2008) to finite supercritical connections. The main distinction from the subcritical analysis is that our argument does not require a separation of masses between irreducible and unrestricted finite connections. The volume bound and a modified coarse-graining construction instead provide stretched-exponential tails for the effective random-walk increments, which suffice to obtain the sharp asymptotics.

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Yacine Aoun, Kamil Khettabi. 2026-10-06. Non-perturbative finite connectivities in supercritical Bernoulli percolation on $\mathbb{Z}^d$ with $d\geq 3$. https://arxiv.org/abs/2610.08682

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