TY - JOUR

T1 - Critical percolation on random regular graphs

AU - Nachmias, Asaf

AU - Peres, Yuval

PY - 2010/3

Y1 - 2010/3

N2 - The behavior of the random graph G(n,p) around the critical probability pc=1/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.

AB - The behavior of the random graph G(n,p) around the critical probability pc=1/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.

KW - Percolation

KW - Random regular graph

UR - http://www.scopus.com/inward/record.url?scp=74349101469&partnerID=8YFLogxK

U2 - 10.1002/rsa.20277

DO - 10.1002/rsa.20277

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AN - SCOPUS:74349101469

SN - 1042-9832

VL - 36

SP - 111

EP - 148

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -