From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

TL;DR

This paper establishes exact mappings between various statistical physics models, linking the steady-state TASEP with open boundaries to a 1D lattice gas model at negative fugacity, revealing deep structural connections.

Contribution

It introduces a novel exact mapping between the steady state of 1D TASEP and 1D lattice gases with negative fugacity, expanding understanding of their interrelations.

Findings

Generating function of TASEP steady state as a quotient of lattice gas partition functions

Representation of TASEP configurations via heaps of pieces

Connection between heaps of pieces and 1D lattice gas partition functions

Abstract

We demonstrate here a series of exact mappings between particular cases of four statistical physics models: equilibrium 1-dimensional lattice gas with nearest-neighbor repulsion, $(1+1)$-dimensional combinatorial heap of pieces, random walks on half-plane and totally asymmetric simple exclusion process (TASEP) in one dimension (1D). In particular, we show that generating function of a steady state of one-dimensional TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of (i) a representation of the steady-state TASEP configurations in terms of $(1+1)$-dimensional heaps of pieces and (ii) a theorem connecting the partition function of $(1+1)$-dimensional heaps of pieces with that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.