Local limits of random graphs

A graph G is a couple G = (V, E) where V denotes the set of vertices of G and V the set of undirected edges. We will assume that the graphs considered are simple, that is they do not have multiple edge nor loop. The degree deg(v) of a vertex v of G is the number of edges attached to v. In the following except explicitly mentioned all the graphs considered are locally finite (no vertex of infinite degree), countable and connected. The graph distance in G is denoted by dG gr(., .). A rooted graph (G,ρ) is a graph G together with a distinguished vertex ρ of G. Two rooted graphs (G,ρ) and (G′,ρ′) are said to be equivalent (G,ρ)' (G′,ρ′) if there is a graph homomorphism G→ G′ that maps ρ to ρ′. Following [4], we define a pseudo-metric on the set of all locally finite connected rooted graphs by