Mergin Percolation on Zd and classical random graphs: Phase transition

Tatyana s. Turova and Thomas Vallier

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
We study a random graph model which is a superposition of the bond percolation model on Zd with probability p of an edge, 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 similar to the classical random graph, it is of the second order. We also find the scaled size of the largest connected component above the phase transition.