The phase transition in site percolation on pseudo-random graphs

We establish the existence of the phase transition in site percolation on pseudo- random d-regular graphs. Let G = (V, E) be an (n, d, λ)-graph, that is, a d-regular graph on n vertices in which all eigenvalues of the adjacency matrix, but the first one, are at most λ in their absolute values. Form a random subset R of V by putting every vertex υ ∈ V into R independently with probability p. Then for any small enough constant ϵ > 0, if (formula presented), then with high probability all connected components of the subgraph of G induced by R are of size at most logarithmic in n, while for (formula presented), if the eigenvalue ratio λ/d is small enough as a function of ϵ, then typically R contains a connected component of size at least (formual presented) and a path of λ/d and a path of length proportional to (formual presented).