Abstract

This talk concerns a  diffusion limit  for  an interacting spin model defined in terms of a multi-component  Markov chain  whose components (spins)  are indexed by vertices of a finite graph. The spins take values in a finite  set of non-negative integers and evolve subject to  a graph based log-linear  interaction.  We show that if the set of possible spin values 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. Although the idea of the proof is similar to those used  for obtaining heavy traffic limits,  some modifications are required due to the presence of interaction.  The talk is based on the joint work with Anatolii Puhal’skii.