Critical percolation on random regular graphs

TL;DR

This paper analyzes the sizes of components in critical percolation on random regular graphs, revealing mean-field behavior, scaling windows, and the distribution of largest components in different regimes.

Contribution

It provides a detailed description of component size distributions at criticality and in near-critical regimes for random regular graphs, including new scaling laws and duality principles.

Findings

Largest components scale as n^{2/3} at criticality

Scaling window width is n^{-1/3} around p_c

Component sizes are concentrated around explicit functions in sub- and supercritical regimes

Abstract

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In particular, we show that there is a scaling window of width n^{-1/3} around p_c in which the sizes of the largest components are roughly n^{2/3} and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))p_c where eps(n)=o(1) but \eps(n)n^{1/3} tends to infinity, the sizes of the largest components are concentrated around an explicit function of n and eps(n) which is of order o(n^{2/3}). In the supercritical regime, i.e. p = (1+\eps(n))p_c where eps(n)=o(1) but eps(n)n^{1/3} tends to infinity, the size of the largest component is concentrated around the value (2d/(d-2))\eps(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.