Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

TL;DR

This paper investigates the growth and isoperimetric properties of infinite clusters in slightly supercritical Bernoulli bond percolation on nonamenable graphs, revealing exponential volume growth and precise estimates near the critical probability.

Contribution

It establishes that infinite clusters exhibit purely exponential volume growth under certain conditions and provides precise growth rate estimates near the critical point, extending understanding of percolation on nonamenable graphs.

Findings

Volume growth of infinite clusters is purely exponential.

Precise estimates for cluster volume near critical probability.

Quenched and annealed growth rates coincide in the supercritical regime.

Abstract

We study the growth and isoperimetry of infinite clusters in slightly supercritical Bernoulli bond percolation on transitive nonamenable graphs under the $L^2$ boundedness condition ($p_c<p_{2\to 2}$). Surprisingly, we find that the volume growth of infinite clusters is always purely exponential (that is, the subexponential corrections to growth are bounded) in the regime $p_c<p<p_{2\to 2}$, even when the ambient graph has unbounded corrections to exponential growth. For $p$ slightly larger than $p_c$, we establish the precise estimates \begin{align*} \mathbf{E}_p \left[ \# B_\mathrm{int}(v,r) \right] &\asymp \left(r \wedge \frac{1}{p-p_c} \right)^{\phantom{2}} e^{\gamma_\mathrm{int}(p) r} \\ \mathbf{E}_p \left[ \# B_\mathrm{int}(v,r) \mid v \leftrightarrow \infty \right] &\asymp \left(r \wedge \frac{1}{p-p_c} \right)^2 e^{\gamma_\mathrm{int}(p) r} \end{align*} for every $v\in V$, $r \geq 0$, and $p_c < p \leq p_c+\delta$, where the growth rate $\gamma_\mathrm{int}(p) = \lim \frac{1}{r} \log \mathbf{E}_p\#B(v,r)$ satisfies $\gamma_\mathrm{int}(p) \asymp p-p_c$. We also prove a percolation analogue of the Kesten-Stigum theorem that holds in the entire supercritical regime and states that the quenched and annealed exponential growth rates of an infinite cluster always coincide. We apply these results together with those of the first paper in this series to prove that the anchored Cheeger constant of every infinite cluster $K$ satisfies \[ \frac{(p-p_c)^2}{\log[1/(p-p_c)]} \preceq \Phi^*(K) \preceq (p-p_c)^2 \] almost surely for every $p_c<p\leq1$.