Slightly supercritical percolation on nonamenable graphs I: The distribution of finite clusters

We study the distribution of finite clusters in slightly supercritical ($p \downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if $G$ is a transitive nonamenable graph satisfying the $L^2$ boundedness condition ($p_c<p_{2\to 2}$) and $K$ denotes the cluster of the origin then there exists $\delta>0$ such that $$\mathbf{P}_p(n \leq |K| < \infty) \asymp n^{-1/2} \exp\left[ -\Theta \Bigl( |p-p_c|^2 n\Bigr) \right]$$ and $$\mathbf{P}_p(r \leq \operatorname{Rad}(K) < \infty) \asymp r^{-1} \exp\left[ -\Theta \Bigl( |p-p_c| r\Bigr) \right]$$ for every $p\in (p_c-\delta,p_c+\delta)$ and $n,r\geq 1$, where all implicit constants depend only on $G$. We deduce in particular that the critical exponents $\gamma'$ and $\Delta'$ describing the rate of growth of the moments of a finite cluster as $p \downarrow p_c$ take their mean-field values of $1$ and $2$ respectively. These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius $\rho<1/2$. In particular, every finitely generated nonamenable 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 $\mathbb{Z}^d$ even for $d$ 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.