Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume

TL;DR

This paper investigates the late-time behavior of height fluctuations in a one-dimensional exclusion process, revealing their relation to extremal values of Brownian bridges and proposing conjectures based on Bethe ansatz results.

Contribution

It introduces a novel reformulation linking KPZ fluctuations to Brownian bridges and conjectures new probabilistic relations for non-intersecting Brownian bridges.

Findings

Height fluctuations relate to extremal Brownian bridge values.

Explicit conjectures for non-intersecting Brownian bridge probabilities.

Comparison with Bethe ansatz yields new asymptotic insights.

Abstract

Height fluctuations are studied in the one-dimensional totally asymmetric simple exclusion process with periodic boundaries, with a focus on how late time relaxation towards the non-equilibrium steady state depends on the initial condition. Using a reformulation of the matrix product representation for the dominant eigenstate, the statistics of the height at large scales is expressed, for arbitrary initial conditions, in terms of extremal values of independent standard Brownian bridges. Comparison with earlier exact Bethe ansatz asymptotics leads to explicit conjectures for some conditional probabilities of non-intersecting Brownian bridges with exponentially distributed distances between the endpoints.