The Benjamini-Schramm nonuniqueness conjecture: p_c < p_u for bond percolation on nonamenable quasi-transitive graphs

Theorem 1.1: for every infinite connected locally finite quasi-transitive graph with positive vertex isoperimetric constant, $\sup_{p<p_c}\|T_p\|_{2\to2}=\|T_{p_c}\|_{2\to2}<\infty$ and $p_c<p_{2\to2}\le p_u$; there are $p_c<p_1<p_2<1$ such that, in the uniform-label coupling, almost surely infinitely many infinite clusters exist simultaneously for all $p\in[p_1,p_2]$. Corollary 1.2 gives the same for every finite symmetric generating set of every nonamenable finitely generated group. Further consequences: the triangle condition at $p_c$, mean-field critical exponents, and exponential decay of connections below $p_{2\to2}$. It concerns bond percolation only; the site-percolation form of the conjecture and the related $p_u<1$ question for one-ended graphs are not addressed. Constants and the interval depend on $G$.

Theorem 1.1: for every infinite connected locally finite quasi-transitive graph with positive vertex isoperimetric constant, $\sup_{p<p_c}\|T_p\|_{2\to2}=\|T_{p_c}\|_{2\to2}<\infty$ and $p_c<p_{2\to2}\le p_u$; there are $p_c<p_1<p_2<1$ such that, in the uniform-label coupling, almost surely infinitely many infinite clusters exist simultaneously for all $p\in[p_1,p_2]$. Corollary 1.2 gives the same for every finite symmetric generating set of every nonamenable finitely generated group. Further consequences: the triangle condition at $p_c$, mean-field critical exponents, and exponential decay of connections below $p_{2\to2}$. It concerns bond percolation only; the site-percolation form of the conjecture and the related $p_u<1$ question for one-ended graphs are not addressed. Constants and the interval depend on $G$.