论文标题
在不合同的图上略有临界渗透II：无限簇的生长和等法
Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters
论文作者
论文摘要
我们研究了在$ l^2 $界面条件下，在略微临界的bernoulli键渗透中，无限簇的生长和等级法（$ p_c <p_c <p_ {2 \ to 2} $）。令人惊讶的是，我们发现无限簇的体积生长始终是纯指数级的（即，即使环境图无限制的对指数增长的校正，即使p_c <p <p_ {2 \ to 2} $，在制度$ p_c <p <p <p_ {2 \ 2} $中始终是纯指数的。对于$ p $略大于$ p_c $，我们建立了精确的估计\ begin {align*} \ mathbf {e} _p \ left [\＃b_ \ mathrm {int}（int}（v，v，v，r）\ right] \ right）^{\ phantom {2}} e^{γ_\ Mathrm {int}（p）r} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \左[ \ \ frac {1} {p-p_c} \ right）^2 e^{γ_\ mathrm {int}（p）r} \ end end {align*} in v $，$ v $，$ r \ geq 0 $，$ v \ in $ v \ geq 0 $，以及$ p_c p_c <p_c p_c p_c p_c+leq p_c+Δ$，pragrip $， \ lim \ frac {1} {r} \ log \ mathbf {e} _p \ #b（v，r）$满足$γ_\ mathrm {int}（p）\ asymp p-p_c $。我们还证明了在整个超临界制度中持有的Kesten-Stigum定理的渗透类似物，并指出，无限集群的淬灭和退火指数增长率始终重合。我们将这些结果与本系列中的第一篇论文一起应用，以证明每个无限群集$ k $的锚cheger常数满足\ [\ frac {（p-p_c）^2} {\ log [1/（p-p_c）} $ p_c <p \ leq1 $。
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^{γ_\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^{γ_\mathrm{int}(p) r} \end{align*} for every $v\in V$, $r \geq 0$, and $p_c < p \leq p_c+δ$, where the growth rate $γ_\mathrm{int}(p) = \lim \frac{1}{r} \log \mathbf{E}_p\#B(v,r)$ satisfies $γ_\mathrm{int}(p) \asymp p-p_c$. 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 \[ \frac{(p-p_c)^2}{\log[1/(p-p_c)]} \preceq Φ^*(K) \preceq (p-p_c)^2 \] almost surely for every $p_c<p\leq1$.
