A Diffusion Limit for Markov Chains with Log-linear Interaction on a Graph

In this paper we establish a diffusion limit for a multivariate continuous time Markov chain whose components are indexed by vertices of a finite graph. The components take values in a common finite set of non-negative integers and evolve subject to a graph based log-linear interaction. We show that if the set of common values of the components expands to the set of all non-negative integers, then a time-scaled and normalised version of the Markov chain converges to a system of interacting Ornstein-Uhlenbeck processes reflected at the origin. This limit is akin to heavy traffic limits in queueing (and our model can be naturally interpreted as a queueing model). Our proof draws on developments in queueing theory and relies on martingale methods.