# Perturbative solution for the spectral gap of the weakly asymmetric exclusion process

## TL;DR

This paper develops a perturbative method using Bethe ansatz to analyze the spectral gap of the weakly asymmetric exclusion process, revealing how Bethe roots behave near the symmetric limit and providing systematic expansions.

## Contribution

It introduces a novel perturbative approach to compute the spectral gap of the weakly asymmetric exclusion process near the symmetric case using Bethe ansatz and functional equations.

## Key findings

- Derived a systematic perturbative expansion of the spectral gap.
- Identified the behavior of Bethe roots near the Fermi sea edge.
- Showed convergence of Bethe roots to zeros of 1+erf(x) after rescaling.

## Abstract

We consider the weakly asymmetric exclusion process with $N=L/2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-\mu/\sqrt{L}$ respectively in the forward and in the backward direction. Using Bethe ansatz, we obtain a systematic perturbative expansion of the spectral gap near $\mu=0$ by solving order by order a simple functional equation. A key point is that when $\mu\to0$, Bethe roots at a distance $1/\sqrt{L}$ from the edge of the Fermi sea should not be considered as a continuum, but converge instead at large $L$ to the complex zeroes of $1+\mathrm{erf}(x)$ after a rescaling by $\sqrt{L}$.