From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation

TL;DR

This paper establishes a novel connection between the sine-Gordon quantum field theory and the KPZ growth equation, linking quantum correlations to stochastic growth phenomena through the non-relativistic limit and Fredholm determinants.

Contribution

It reveals a new relationship between sine-Gordon theory and KPZ dynamics, showing how quantum field correlations relate to stochastic growth distributions.

Findings

Non-relativistic limit of sine-Gordon correlators relates to KPZ height distribution.

The KPZ generating function can be expressed as a Fredholm determinant.

Large time behavior converges to GUE Tracy-Widom distribution.

Abstract

We unveil a remarkable connection between the sine-Gordon quantum field theory and the Kardar-Parisi-Zhang (KPZ) growth equation. We find that the non-relativistic limit of the two point correlation function of the sine-Gordon theory is related to the generating function of the height distribution of the KPZ field with droplet initial conditions, i.e. the directed polymer free energy with two endpoints fixed. As shown recently, the latter can be expressed as a Fredholm determinant which in the large time separation limit converges to the GUE Tracy-Widom cumulative distribution. Possible applications and extensions are discussed.