Recursive Packing Bounds for Supercritical Disconnection in Bernoulli Site Percolation

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.