The Benjamini-Schramm criticality conjecture for bond percolation on quasi-transitive graphs

Claims that for every infinite connected locally finite quasi-transitive graph with $p_c<1$, Bernoulli bond percolation at $p_c$ has almost surely no infinite cluster. Exponential growth is Hutchcroft's theorem; the new work covers all subexponential growth, split into superpolynomial growth (a two-cluster entropy argument) and growth polynomial along a sequence of radii (a nilpotent-quotient corridor exploration). It includes $\theta(p_c)=0$ on $\mathbb Z^d$ for every $d\ge2$. The companion proves critical bond and site nonpercolation on $\mathbb Z^3$. It does NOT settle the site form of the conjecture on general quasi-transitive graphs, and gives no critical exponents or quantitative decay.

Claims that for every infinite connected locally finite quasi-transitive graph with $p_c<1$, Bernoulli bond percolation at $p_c$ has almost surely no infinite cluster. Exponential growth is Hutchcroft's theorem; the new work covers all subexponential growth, split into superpolynomial growth (a two-cluster entropy argument) and growth polynomial along a sequence of radii (a nilpotent-quotient corridor exploration). It includes $\theta(p_c)=0$ on $\mathbb Z^d$ for every $d\ge2$. The companion proves critical bond and site nonpercolation on $\mathbb Z^3$. It does NOT settle the site form of the conjecture on general quasi-transitive graphs, and gives no critical exponents or quantitative decay.