Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

TL;DR

This paper develops conjectures for the asymptotic behavior of the ASEP eigenstates near the stationary state, verified through Bethe ansatz numerics, and derives an exact spectral gap expression for weak asymmetry.

Contribution

It introduces precise conjectures for ASEP eigenstate asymptotics at finite density and provides an exact spectral gap formula for weak asymmetry using advanced extrapolation methods.

Findings

Conjectures for ASEP eigenstate asymptotics near stationary state.

High-precision verification using Bethe ansatz numerics.

Exact spectral gap expression for weak asymmetry up to 10th order.

Abstract

The one-dimensional asymmetric simple exclusion process (ASEP), where $N$ hard-core particles hop forward with rate $1$ and backward with rate $q<1$, is considered on a periodic lattice of $L$ site. Using KPZ universality and previous results for the totally asymmetric model $q=0$, precise conjectures are formulated for asymptotics at finite density $\rho=N/L$ of ASEP eigenstates close to the stationary state. The conjectures are checked with high precision using extrapolation methods on finite size Bethe ansatz numerics. For weak asymmetry $1-q\sim1/\sqrt{L}$, double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to $10$-th order in the asymmetry.