Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature

TL;DR

This paper derives an exact expression for the short-time height distribution in the 1D KPZ equation with droplet geometry, revealing Gaussian center and asymmetric tails, and connects these results to edge fermion fluctuations at high temperature.

Contribution

It provides the exact rate function for the height distribution in the KPZ equation at short times, revealing detailed tail behaviors and linking to fermionic edge fluctuations.

Findings

Exact rate function for height distribution derived

Gaussian behavior at the center with asymmetric tails characterized

Implications for fermion edge fluctuations at high temperature

Abstract

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions in curved (or droplet) geometry. We show that for short time $t$, the probability distribution $P(H,t)$ of the height $H$ at a given point $x$ takes the scaling form $P(H,t) \sim \exp{\left(-\Phi_{\rm drop}(H)/\sqrt{t} \right)}$ where the rate function $\Phi_{\rm drop}(H)$ is computed exactly. While it is Gaussian in the center, i.e., for small $H$, the PDF has highly asymmetric non-Gaussian tails which we characterize in detail. This function $\Phi_{\rm drop}(H)$ is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite size models belonging to the KPZ universality class. Thanks to a recently discovered connection between KPZ and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full couting statistics at the edge.