A law of large numbers and large deviations for interacting diffusions on Erdős–Rényi graphs

Abstract

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős–Rényi graph with parameter [Formula: see text], where [Formula: see text] is the size of the graph (i.e. the number of particles). If [Formula: see text], the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as [Formula: see text] to the solution of a PDE: a McKean–Vlasov (or Fokker–Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős–Rényi graphs with [Formula: see text], and properly rescaling the interaction to account for the dilution introduced by [Formula: see text]. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results — Law of Large Numbers and Large Deviation Principle — assuming only the convergence of the empirical measure of the initial condition.