Mixing times and spectra of non-equilibrium symmetric exclusion processes on general graphs

Abstract

The symmetric exclusion process (SEP) is a classical model of interacting particles on a graph, in which multiple particles execute random walks subject to the constraint that no two particles may occupy the same vertex. The version of the process in which the number of particles is fixed, is by now very well understood, through the study of reversible Markov chains. In this paper, we study the non-equilibrium version of the SEP, in which some vertices of the graph interact with external heat baths, held at different temperatures, that mediate transport through the graph. The resulting steady-state distribution is still the stationary distribution of a natural Markov chain, but the chain is no longer reversible. This means that even the steady-state distribution is very hard to describe, and indeed is the subject of a rich literature, which has focused almost exclusively on graphs which are finite subsets of the lattice $Z^d$. Very little is known about other important quantities, notably the mixing time. We derive various quantitative results about the non-equilibrium SEP for arbitrary graphs with arbitrary heat baths. Our main result is a bound on the mixing time in terms of the worst-case hitting time of a single particle to any of the heat baths; this bound is tight up to two logarithmic factors in the size of the graph. We then use this result to derive a simple algorithm, running in time roughly $n^{O(k)}$, that computes the steady-state joint occupation distribution for any set of $k$ vertices; since the steady-state distribution is very elusive, this provides a potentially useful tool to study its $k$-wise correlations for small values of $k$. Finally, we prove that the spectrum of the Markov chain associated with the non-reversible dynamics for the SEP is independent of the heat bath temperatures; thus in particular the spectrum is always real.