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

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 $\mu \to 0$, 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 $\sqrt{L}$.

28 pages, 5 figures

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Cited by in corpus (6)

• Riemann surfaces for KPZ with periodic boundaries
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Equivalent T-Q relations and exact results for the open TASEP
• KPZ fluctuations in finite volume
• Random sparse generators of Markovian evolution and their spectral properties
• From the Riemann surface of TASEP to ASEP