Integrability and exact large deviations of the weakly-asymmetric exclusion process

TL;DR

This paper derives exact formulas for large deviations in the weakly asymmetric exclusion process, revealing its integrability and connecting it to the KPZ universality class and classical spin chains.

Contribution

It provides the first explicit Lax pairs for the MFT of WASEP, demonstrating its classical integrability and mapping it to a complex Landau-Lifshitz spin chain.

Findings

Exact cumulant generating functions and large deviation rate functions derived.

Shows crossover behavior between SSEP and KPZ regimes.

Establishes integrability of the MFT for WASEP and related models.

Abstract

The weakly asymmetric exclusion process (WASEP) in one dimension is a paradigmatic system of interacting particles described by the macroscopic fluctuation theory (MFT) in the presence of driving. We consider an initial condition with densities $\rho_1,\rho_2$ on either side of the origin, so that for $\rho_1=\rho_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulae for the cumulant generating functions, and large deviation rate functions of the time-integrated current and the position of a tracer. As the asymmetry/driving is increased, these describe the crossover between the symmetric exclusion process (SSEP) and the weak noise regime of the Kardar-Parisi-Zhang (KPZ) equation: we recover the two limits and describe the crossover from the WASEP cubic tail to the $5/2$ and $3/2$ KPZ tail exponents. Finally, we show that the MFT of the WASEP is classically integrable, by exhibiting the explicit Lax pairs, which are obtained through a novel mapping between the MFT of the WASEP and a complex extension of the classical anisotropic Landau-Lifshitz spin chain. This shows integrability of all MFTs of asymmetric models with quadratic mobility as well as their dual versions.