# Combinatorial mappings of exclusion processes

## TL;DR

This paper reviews combinatorial interpretations of stationary states in exclusion processes, highlighting mappings to paths and permutations that simplify analysis and connect physical properties with combinatorial structures.

## Contribution

It consolidates existing results and introduces new findings on combinatorial mappings of exclusion processes, bridging statistical physics and combinatorics.

## Key findings

- Mappings to random walks and permutations simplify steady-state analysis
- Configuration weights often have simple Boltzmann-like forms in extended spaces
- Physical properties like entropy relate to partitions in combinatorial mappings

## Abstract

We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties. Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state - such as the entropy - can be interpreted in terms of how this larger state space has been partitioned. These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.