On limits of sparse random graphs

We present a notion of convergence for sequences of finite graphs {Gn} that can be seen as a generalization of the Benjamini-Schramm convergence notion for bounded degree graphs, regarding the distribution of r-neighbourhoods of the vertices, and the left-convergence notion for dense graphs, regarding, given any finite graph F, the limit of the probabilities that a random map from V(F) to V(Gn) is a graph homomorphism. Furthermore, this presented convergence notion allows us to define, for each p(n) and with high probability, a limit for a sequence of Erdős-Renyi random graphs with Gn∼G(n,p(n)).