Abstract
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 marked with the initial conditions. In addition, we show that the global empirical measure converges to a nonrandom limit for a large class of graph sequences including sparse Erdős–Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.
Funding Statement
D. Lacker was partially supported by the Air Force Office of Scientific Research (AFOSR) Grant FA9550-19-1-0291.
K. Ramanan was partially supported by the Army Research Office (ARO) Grant W911NF2010133, a Simons Fellowship, and the Office of Naval Research under the Vannevar Bush Faculty Fellowship N0014-21-1-2887.
Acknowledgments
We thank an anonymous referee for suggesting the more direct and general argument presented in the proof of Proposition 7.8. In the case of an amenable group G, the mean ergodic theorem states that the ergodicity of x is equivalent to convergence of the empirical fields for any Følner sequence (see [17], Chapter 8, for definitions). In a previous version of the paper, we used this along with Proposition 7.3 to prove Proposition 7.8 in the amenable case.
Citation
Daniel Lacker. Kavita Ramanan. Ruoyu Wu. "Local weak convergence for sparse networks of interacting processes." Ann. Appl. Probab. 33 (2) 843 - 888, April 2023. https://doi.org/10.1214/22-AAP1830
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