Mean-field conditions for percolation on finite graphs

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Abstract

Let {Gn} be a sequence of finite transitive graphs with vertex degree d=d(n) and |Gn| =n. Denote by pt(v, v) the return probability after t steps of the non-backtracking random walk on Gn. We show that if pt(v, v) has quasi-random properties, then critical bond-percolation on Gn behaves as it would on a random graph. More precisely, if then the size of the largest component in p-bond-percolation with is roughly n2/3. 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. A consequence of our theorems is that if {Gn} is a transitive expander family with girth at least then {Gn} has the above scaling window around. 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.

Original languageEnglish
Pages (from-to)1171-1194
Number of pages24
JournalGeometric and Functional Analysis
Volume19
Issue number4
DOIs
StatePublished - Dec 2009
Externally publishedYes

Keywords

  • Giant component
  • Percolation
  • Random graphs
  • Scaling window

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