2007/07/19 by Asaf Nachmias, Yuval Peres, Nachmias, Asaf +1 · 1 citation
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Complex Network Analysis Techniques #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.0707.2839
We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d ≥ 3 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-1/3 around pc in which the sizes of the largest components are roughly n2/3 and we describe their limiting joint distribution. We also show that for the subcritical regime, i.e. p = (1-eps(n))pc where eps(n)=o(1) but \eps(n)n1/3 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(n2/3). In the supercritical regime, i.e. p = (1+\eps(n))pc where eps(n)=o(1) but eps(n)n1/3 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.