LOCAL WEAK CONVERGENCE FOR SPARSE NETWORKS OF INTERACTING PROCESSES

• 成果类型: Article
• 署名作者: Lacker, Daniel; Ramanan, Kavita; Wu, Ruoyu
• 署名单位: Columbia University; Brown University; Iowa State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1830
• 发表日期: 2023
• 页码: 643-688
• 关键词: Random graphs; DIFFUSIONS; models; systems; limit

摘要

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 conver-gence 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 Erdos-Renyi graphs and configuration models, whereas the empirical measure of the connected com-ponent 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 de-scriptions 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.