Point-interacting Brownian motions in the KPZ universality class

TL;DR

This paper studies a class of interacting Brownian motions with asymmetry, deriving exact formulas and connecting their large-scale behavior to the KPZ universality class, highlighting novel analytical techniques.

Contribution

It introduces a Bethe ansatz formula and self-duality for asymmetric Brownian motion chains, linking them to KPZ universality through asymptotic analysis.

Findings

Bethe ansatz formula for transition probabilities

Fredholm determinant representation for particle counting

Asymptotic link to KPZ universality class

Abstract

We discuss chains of interacting Brownian motions. Their time reversal invariance is broken because of asymmetry in the interaction strength between left and right neighbor. In the limit of a very steep and short range potential one arrives at Brownian motions with oblique reflections. For this model we prove a Bethe ansatz formula for the transition probability and self-duality. In case of half-Poisson initial data, duality is used to arrive at a Fredholm determinant for the generating function of the number of particles to the left of some reference point at any positive time. A formal asymptotics for this determinant establishes the link to the Kardar-Parisi-Zhang universality class.