Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

Abstract

We study the distribution of finite clusters in slightly supercritical ( p ↓ p c $p \downarrow p_c$ ) Bernoulli bond percolation on transitive non-amenable graphs, proving in particular that if G $G$ is a transitive non-amenable graph satisfying the L 2 $L^2$ boundedness condition ( p c < p 2 → 2 $p_c<p_{2\rightarrow 2}$ ) and K $K$ denotes the cluster of the origin and then there exists δ > 0 $\delta >0$ such that if p ∈ ( p c − δ , p c + δ ) $p\in (p_c-\delta ,p_c+\delta )$ , then P p ( n ⩽ | K | < ∞ ) ≍ n − 1 / 2 exp − Θ | p − p c | 2 n \begin{align*} \hskip1.5pc\mathbf {P}_p(n \leqslant |K| < \infty ) &\asymp n^{-1/2} \,{\exp} {\left[ -\Theta {\left(|p-p_c|^2 n\right)} \right]}\hskip-1.5pc \end{align*} and P p ( r ⩽ Rad ( K ) < ∞ ) ≍ r − 1 exp [ − Θ ( | p − p c | r ) ] \begin{align*} \hskip1.5pc\mathbf {P}_p(r \leqslant \operatorname{Rad}(K) < \infty ) &\asymp r^{-1} \exp {[ -\Theta (|p-p_c| r) ]}\hskip-1.5pc \end{align*} for every n , r ⩾ 1 $n,r\geqslant 1$ , where all implicit constants depend only on G $G$ . We deduce in particular that the critical exponents γ ′ $\gamma ^{\prime }$ and Δ ′ $\Delta ^{\prime }$ describing the rate of growth of the moments of a finite cluster as p ↓ p c $p \downarrow p_c$ take their mean-field values of 1 and 2, respectively. These results apply in particular to Cayley graphs of non-elementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ < 1 / 2 $\rho <1/2$ . In particular, every finitely generated non-amenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on Z d $\mathbb {Z}^d$ even for d $d$ very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.