Restricted permutations for the simple exclusion process in discrete time over graphs

TL;DR

This paper introduces a discrete-time version of the simple exclusion process on graphs using restricted permutations, providing a new formalism and simulation method for studying non-equilibrium particle systems.

Contribution

It defines a novel discrete-time exclusion process on graphs via restricted permutations and offers an efficient importance sampling algorithm for simulation.

Findings

The process approaches stationarity with algebraic decay.

The decay exponent varies between 1 and 2 depending on graph matches.

The approach is applicable to loop-augmented Bollobás-Chung graphs.

Abstract

Exclusion processes became paradigmatic models of nonequilibrium interacting particle systems of wide range applicability both across the natural and the applied, social and technological sciences. Usually they are defined as a continuous-time stochastic process, but in many situations it would be desirable to have a discrete-time version of them. There is no generally applicable formalism for exclusion processes in discrete-time. In this paper we define the symmetric simple exclusion process in discrete time over graphs by means of restricted permutations over the labels of the vertices of the graphs and describe a straightforward sequential importance sampling algorithm to simulate the process. We investigate the approach to stationarity of the process over loop-augmented Bollob\'as-Chung "cycle-with-matches" graphs. In all cases the approach is algebraic with an exponent varying between $1$ and $2$ depending on the number of matches.