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 $${\mathbf{P}}_p(n\le \mathrm{\mid }K\mathrm{\mid }<\mathrm{\infty })\asymp n^{-1\mathrm{/}2}\exp [-\mathrm{\Theta }(\mathrm{\mid }p-p_c{\mathrm{\mid }}^{2}n\left)\right]$$ and $${\mathbf{P}}_p(r\le \mathrm{Rad}(K)<\mathrm{\infty })\asymp r^{-1}\exp [-\mathrm{\Theta }(\mathrm{\mid }p-p_c\mathrm{\mid }r\left)\right]$$ 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 ${\gamma }^{\mathrm{\prime }}$ and ${\mathrm{\Delta }}^{\mathrm{\prime }}$ 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 $\rho <1\mathrm{/}2$. 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.