Sharpness and locality for percolation on finite transitive graphs

TL;DR

This paper investigates the phase transition behavior of percolation on finite transitive graphs, establishing conditions under which the transition is sharp and linking it to geometric limits of the graphs.

Contribution

It proves that percolation on such graphs exhibits a sharp phase transition unless the graphs converge to a circle in the Gromov-Hausdorff sense, connecting geometric limits to percolation behavior.

Findings

Percolation has a sharp phase transition unless graphs converge to a circle.

Critical points for giant cluster emergence match those in the Benjamini-Schramm limit.

Identifies geometric conditions affecting percolation phase transition.

Abstract

Let $(G_n) = \left((V_n,E_n)\right)$ be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that $\lvert V_n \rvert \to \infty$ as $n \to \infty$. We say that percolation on $G_n$ has a sharp phase transition (as $n \to \infty$) if, as the percolation parameter crosses some critical point, the number of vertices contained in the largest percolation cluster jumps from logarithmic to linear order with high probability. We prove that percolation on $G_n$ has a sharp phase transition unless, after passing to a subsequence, the rescaled graph-metric on $G_n$ (rapidly) converges to the unit circle with respect to the Gromov-Hausdorff metric. We deduce that under the same hypothesis, the critical point for the emergence of a giant (i.e. linear-sized) cluster in $G_n$ coincides with the critical point for the emergence of an infinite cluster in the Benjamini-Schramm limit of $(G_n)$, when this limit exists.