Another look at the Bal\'azs-Quastel-Sepp\"al\"ainen theorem

TL;DR

This paper provides a new proof of the KPZ equation's variance growth rate using directed polymers, confirming a physics prediction and avoiding previous discrete approximation methods.

Contribution

It introduces a novel proof technique connecting directed polymers to the KPZ variance, bypassing discrete approximations and confirming a key physics prediction.

Findings

Variance of KPZ solution grows as t^{2/3}

Annealed density of directed polymer equals the two-point covariance

Confirmed physics prediction about the connection between polymers and KPZ

Abstract

We study the KPZ equation with a $1+1-$dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of \cite{bqs}, i.e., the variance of the solution at time $t$ is of order $t^{2/3}$. Instead of using a discrete approximation through the exclusion process and the second class particle, we utilize the connection to directed polymers in a random environment. Along the way, we show the annealed density of the stationary continuum directed polymer equals to the two-point covariance function of the stationary stochastic Burgers equation, confirming the physics prediction in \cite{MT}.