April 2023 Local weak convergence for sparse networks of interacting processes
Daniel Lacker, Kavita Ramanan, Ruoyu Wu
Author Affiliations +
Ann. Appl. Probab. 33(2): 843-888 (April 2023). DOI: 10.1214/22-AAP1830


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.


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 1|A n| gAnδgxL(x) for any Følner sequence (An) (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.


Download 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


Received: 1 May 2021; Revised: 1 December 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692278
MathSciNet: MR4564415
Digital Object Identifier: 10.1214/22-AAP1830

Primary: 60J05 , 60J60 , 60J80 , 60K35
Secondary: 60B10 , 60F17 , 82C22

Keywords: configuration model , discrete-time Markov chains , Erdős–Rényi graphs , Gibbs measures , Interacting diffusions , Local weak convergence , Markov random fields , mean-field limits , nonlinear Markov processes , Probabilistic cellular automata , Random graphs , sparse graphs , unimodularity

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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