Mean-Field Conditions for Percolation on Finite Graphs

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 behaves as it would on a random graph. More precisely, if $$\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$$then the size of the largest component in p-bond-percolation with \({p =\frac{1+O(n^{-1/3})}{d-1}}\) is roughly n 2/3. In Physics jargon, this condition implies that there exists a scaling window with a mean-field width of n −1/3 around the critical probability \({p_c =\frac{1}{d-1}}\).A consequence of our theorems is that if {G n } is a transitive expander family with girth at least \({\left( {\frac{2}{3} + \epsilon} \right) {\rm log}_{d - 1}n}\) then {G n } has the above scaling window around \({p_c =\frac{1}{d-1}}\). In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak [LuPS] has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.