Local weak convergence and propagation of ergodicity for sparse networks of interacting processes.

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph. In addition, we show that the global empirical measure converges to a non-random limit, whereas for a large class of graph sequences including sparse Erd\"{o}s-R\'{e}nyi graphs and configuration models, the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Finally, on a lattice (or more generally an amenable Cayley graph), we show that if the initial configuration of the particle system is a stationary ergodic random field, then so is the configuration of particle trajectories up to any fixed time, a phenomenon we refer to as "propagation of ergodicity". Along the way, we develop some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new.