Resolvent of large random graphs

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdös-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. MSC-class: 05C80, 15A52 (primary), 47A10 (secondary). 1 Definition and Main Results 1.1 Convergence of the spectral measure of random graphs We denote a (multi-)graph G with vertex set V and undirected edge set E by G = (V,E). The degree of the vertex v ∈ V in G is deg(G, v). In this paper, a network is a graph G = (V,E) together with a complete separable space H called the mark space and maps from V to H. When needed, we use the following notation: G will denote a graph and G a network with underlying graph G. A rooted network (G, o) is a network G with a distinguished vertex o of G, called the root. A rooted isomorphism of rooted networks is an isomorphism of the underlying networks that takes the root of one to the root of the other. [G, o] will denote the class of rooted networks that are rooted-isomorphic to (G, o). We shall use the following notion introduced by Benjamini and Schramm [5] and Aldous and Steele [2]. Let G∗ (respectively G∗) denote the set of rooted isomorphism classes of rooted connected locally finite networks (respectively graphs). Define a metric on G∗ by letting the distance between (G1, o1) and (G2, o2) be 1/α, where α is the supremum of those r > 0 such that there is some rooted isomorphism of the balls of (graph-distance) radius ⌊r⌋ around the roots of Gi such that each mark has distance less than 1/r. We define the same metric on G∗, by considering that a graph is a network with a constant Institut de Mathématiques Université de Toulouse & CNRS France. Email: [email protected] INRIA-ENS France. Email: [email protected]