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The phase transition in site percolation on pseudo-random graphs

unpublished

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 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

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