Critical percolation on random regular graphs

Abstract

AbstractThe behavior of the random graph G(n,p) around the critical probability pc = $ {1 \over n} $ is well understood. When p = (1 + O(n1/3))pc the components are roughly of size n2/3 and converge, when scaled by n−2/3, to excursion lengths of a Brownian motion with parabolic drift. In particular, in this regime, they are not concentrated. When p = (1 ‐ ϵ(n))pc with ϵ(n)n1/3 →∞ (the subcritical regime) the largest component is concentrated around 2ϵ−2 log(ϵ3n). When p = (1 + ϵ(n))pc with ϵ(n)n1/3 →∞ (the supercritical regime), the largest component is concentrated around 2ϵn and a duality principle holds: other component sizes are distributed as in the subcritical regime.Itai Benjamini asked whether the same phenomenon occurs in a random d‐regular graph. Some results in this direction were obtained by (Pittel, Ann probab 36 (2008) 1359–1389). In this work, we give a complete affirmative answer, showing that the same limiting behavior (with suitable d dependent factors in the non‐critical regimes) extends to random d‐regular graphs. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010