Recursive Packing Bounds for Supercritical Disconnection in Bernoulli Site Percolation

TL;DR

This paper introduces a recursive packing number to derive quantitative upper bounds on the probability of supercritical disconnection in Bernoulli site percolation on infinite graphs, providing explicit estimates applicable to various graph structures.

Contribution

It develops a novel recursive packing bound for disconnection probabilities in supercritical Bernoulli site percolation, connecting local witness counts to global disconnection estimates.

Findings

Derived an explicit upper bound for disconnection probability using the packing number.

Showed the packing number is explicit for certain graph families like ray-homogeneous trees.

Connected the packing number to known critical parameters, enabling practical estimates.

Abstract

For Bernoulli site percolation on an infinite, connected, locally finite graph $G=(V,E)$, we obtain quantitative upper bounds on the supercritical disconnection probability \[ \mathbb{P}_p(S\nleftrightarrow\infty) \] for arbitrary finite or infinite sets $S\subset V$ and all $p>p^{\mathrm{site}}_c(G)$. The key quantity is a recursive packing number $\mathbf{PK}_{p,\eps,c}(S)$. It is the maximal number of vertices that can be extracted from $S$ so that, after deleting witness balls around the previously chosen vertices, each selected vertex still connects to infinity with probability at least $c$, while its failure to connect to infinity is already detected, up to a factor $1+\eps$, by failure to reach the inner boundary of its witness ball. Thus $\mathbf{PK}_{p,\eps,c}(S)$ counts essentially independent local witnesses for the global event $\{S\nleftrightarrow\infty\}$. We prove the structural estimate \[ \mathbb{P}_p(S\nleftrightarrow\infty) \le \frac{\eps(1-c)}{c} +(1-c)^{\mathbf{PK}_{p,\eps,c}(S)}. \] Combining this bound with the local functional characterization of $p^{\mathrm{site}}_c(G)$ from \cite{ZL24} yields an explicit supercritical estimate valid on every infinite, connected, locally finite graph. We also illustrate the packing number on ray-homogeneous trees. In particular, sparse finite subsets of a distinguished ray have packing number equal to their cardinality, both for regular trees and for a non-regular decorated spine. This shows that the packing number is explicit on concrete graph families.