In this talk, we present recent results on the extreme eigenvalues of the adjacency matrix of Erd?s-Rnyi graphs. The Erd?s-Rnyi graph G has N vertices and any two vertices are connected with probability p, independently of other edges.

If p is large then the adjacency matrix A of G behaves like a Wigner random matrix and has the semicircle law on [-2,2] as limiting eigenvalue density. Moreover, the extreme eigenvalues converge to -2 and 2, respectively.

If p is small then, however, A has many eigenvalues outside [-2,2].

Recently, the critical value of p for this transition has been determined and a precise connection between the large degrees of G and the extreme eigenvalues of A has been established.

This is joint work with Raphael Ducatez and Antti Knowles.

Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria www.ista.ac.at © ISTA 2024