Mean-field conditions for percolation on finite graphs

TL;DR

This paper establishes conditions under which finite transitive graphs exhibit percolation behavior similar to random graphs, particularly identifying a scaling window of width n^{-1/3} for critical bond-percolation.

Contribution

It introduces quasi-random conditions on return probabilities that imply random-graph-like percolation scaling in finite transitive graphs.

Findings

Critical percolation on certain graphs has a scaling window of width n^{-1/3}.

Expander graphs with high girth exhibit percolation behavior similar to random graphs.

Ramanujan graphs demonstrate the predicted percolation scaling window.

Abstract

Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n) and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random properties, then critical bond-percolation on G_n has a scaling window of width n^{-1/3}, as it would on a random graph. A consequence of our theorems is that if G_n is a transitive expander family with girth at least (2/3 + eps) \log_{d-1} n, then the size of the largest component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is roughly n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.