Mergin Percolation on Z^{d }and classical random graphs: Phase transition
Tatyana s. Turova and Thomas Vallier
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2006
ISSN 14039338

Abstract:

We study a random graph model which is a superposition of the bond percolation
model on Z^{d} 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 socalled
"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 Z^{d}. 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.





