Critical percolation on random regular graphs

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d is fixed and n grows. We prove mean-field behavior around the critical probability pc = 1 d−1 . In particular, we show that there is a scaling window of width n around pc 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 − ε(n))pc where ε(n) = o(1) but ε(n)n → ∞, the sizes of the largest components are concentrated around an explicit function of n and ε(n) which is of order o(n). In the supercritical regime, i.e. p = (1 + ε(n))pc where ε(n) = o(1) but ε(n)n → ∞, the size of the largest component is concentrated around the value 2d d−2ε(n)n and a duality principle holds: other component sizes are distributed as in the subcritical regime.