Merging percolation on Zd and classical random graphs: Phase transition

: We study a random graph model which is a superposition of bond percolation on Zd with parameter p, and a classical random graph G(n, c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called “rank 1 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < pc, where pc = pc(d) is the critical probability for the bond percolation on Zd . The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 36, 185–217, 2010

• Type: Journal
• Year: 2010
• Where: RSA
• Authors: Tatyana S. Turova, Thomas Vallier
• Keywords: Classical Random Graph, Inhomogeneous Random Graphs, Random Graphs, RSA 2010