Merging percolation on $Z^d$ and classical random graphs: Phase transition

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 {\it homogeneous} random graph, has a natural relation to the so-called "rank 1 case" of {\it 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 $c\geq 0$ and $0 \leq p