Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process

TL;DR

This paper analyzes the large system size asymptotics of Bethe eigenstate norms in the periodic TASEP, revealing they behave like the exponential of a scalar free field's action, relevant for understanding KPZ relaxation.

Contribution

It provides the first detailed asymptotic analysis of Bethe eigenstate normalizations in TASEP, connecting them to scalar free field actions in the large system limit.

Findings

Normalization asymptotics match exponential of scalar free field action

Asymptotics derived using Euler-Maclaurin formula with singularities

Results applicable to relaxation dynamics on KPZ time scale

Abstract

The normalization of Bethe eigenstates for the totally asymmetric simple exclusion process on a ring of $L$ sites is studied, in the large $L$ limit with finite density of particles, for all the eigenstates responsible for the relaxation to the stationary state on the KPZ time scale $T\sim L^{3/2}$. In this regime, the normalization is found to be essentially equal to the exponential of the action of a scalar free field. The large $L$ asymptotics is obtained using the Euler-Maclaurin formula for summations on segments, rectangles and triangles, with various singularities at the borders of the summation range.