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.
- Random regular graph