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 ǫ > 0, if p = 1−ǫ 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 =
More precisely REF studied site percolation on (n, d, λ) graphs.
The phase transition in site percolation on pseudo-random graphs
arXiv: Combinatorics
Mathematics Computer Science