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

TL;DR

The paper studies the one-dimensional weakly asymmetric exclusion process (WASEP) within the macroscopic fluctuation theory (MFT) framework for a two-sided Bernoulli initial condition, deriving exact cumulant generating functions and large deviation rate functions for the time-integrated current $J(X,T)$ and the tracer position $Y_T$. It analyzes the crossover as the asymmetry is scaled as $1/ sqrt{T}$, connecting to the SSEP and the weak-noise KPZ regime, recovering cubic tails in the current distribution and the KPZ lower-tail exponents $5/2$ and $3/2$. A central result is the classical integrability of the MFT for WASEP, achieved by constructing explicit Lax pairs via a mapping to a complex extension of the anisotropic Landau-Lifshitz spin chain, implying integrability for all quadratic-mobility MFTs and their duals. The paper employs two complementary frameworks: an exact Fredholm-determinant approach and an explicit Lax-integrability construction, yielding high-order cumulants and tracer statistics with consistency checks against SSEP and KPZ limits. It also extends the integrable structure to related quadratic MFT models (WASIP, WAKMP) and dual versions, underscoring the broad applicability of the integrable MFT framework to driven diffusive systems with weak asymmetry.

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 $ρ_1,ρ_2$ on either side of the origin, so that for $ρ_1=ρ_2$ 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.

Figures (2)

• Figure SS.2: Numerical solution of the soliton rapidity equation \ref{['eq:soliton-rapidity-equation']} in different regimes. (a) top$T=1$, $w=0.2$, $\nu=1$ where the physical branch of $\kappa(u)$ (i.e., the one departing to the right from $\kappa(u_c)=0$) is single-valued, (b) bottom-left$T=1$, $w=1.3w_c$, $\nu=2.3$ where the physical branch is single-valued, (c) bottom-right$T=1$, $w=0.5w_c$, $\nu=2.3$ above the transition where the physical branch is tri-valued between $u_{c1}$ and $u_{c2}$. We will not study this case in detail in this paper.
• Figure SS.3: Plot of $\Phi(J)$ obtained numerically parametrically for the values $X=0$, $T=1$, $\nu=1$ and various densities $\varrho_1=1-\varrho_2=\{0.6,0.95,1\}$. In the right plots representing $\Phi'(J)$ the blue curve represents the main branch and the orange curve represents the second branch where the continuation is introduced.

Theorems & Definitions (32)

• Remark G.1
• Remark H.1
• Remark H.2
• Remark I.1
• Remark J.1
• Remark J.2
• Remark K.1
• Remark K.2
• Remark K.3
• Remark K.4