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 ${\varrho }_1,{\varrho }_2$ on either side of the origin, so that for ${\varrho }_1={\varrho }_2$ the gas is stationary. Starting from the microscopic description, we obtain exact formulas 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 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.

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• Exclusion processes
• Asymmetric simple exclusion process
• Integrable systems
• Kardar–Parisi–Zhang equation
• Landau-Lifshitz model
• Large deviation & rare event statistics