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

We consider the weakly asymmetric exclusion process with $N=L\mathrm{/}2$ particles on a periodic lattice of $L$ sites, and hopping rates $1$ and $q=1-\mu \mathrm{/}\sqrt{L}$ 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 $\mu \to 0$, Bethe roots at a distance $1\mathrm{/}\sqrt{L}$ 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 $1+\mathrm{e}\mathrm{r}\mathrm{f}(x)$ after a rescaling by $\sqrt{L}$.