  We consider a discrete-time system of n coupled random vectors, a.k.a.
interacting particles. The dynamics involve a vanishing step size, some random
centered perturbations, and a mean vector field which induces the coupling
between the particles. We study the doubly asymptotic regime where both the
number of iterations and the number n of particles tend to infinity, without
any constraint on the relative rates of convergence of these two parameters. We
establish that the empirical measure of the interpolated trajectories of the
particles converges in probability, in an ergodic sense, to the set of
recurrent Mc-Kean-Vlasov distributions. A first application example is the
granular media equation, where the particles are shown to converge to a
critical point of the Helmholtz energy. A second example is the convergence of
stochastic gradient descent to the global minimizer of the risk, in a wide
two-layer neural networks using random features.
