The Phase Transition in Site Percolation on Pseudo-Random Graphs

We establish the existence of the phase transition in site percolation on pseudo-random dregular 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 v ∈ V into R independently with probability p. Then for any small enough constant > 0, if p = 1− d , 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 p = 1+ d , if the eigenvalue ratio λ/d is small enough as a function of , then typically R contains a connected component of size at least n d and a path of length proportional to n d .