Merging Percolation on Zd and Classical Random Graphs: Phase Transition

We study a random graph model which is a superposition of bond percolation on Z(d) 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 I 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 < p(c), where p(c) = p(c)(d) is the critical probability for the bond percolation on Z(d). 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 supercirtical 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. (C) 2009 Wiley Periodicals, Inc. Random Struct, Alg., 36, 185-217, 2010