The giant component threshold for random regular graphs with edge faults
Theoretical Computer Science
Random subgraphs of finite graphs: I. The scaling window under the triangle condition
Random Structures & Algorithms
The evolution of the cover time
Combinatorics, Probability and Computing
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The 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