Dynamical phase transition in large deviation statistics of the Kardar-Parisi-Zhang equation

TL;DR

This paper analyzes the short-time large deviation statistics of the KPZ equation surface height, revealing a second-order phase transition and connecting tail behaviors to known distributions.

Contribution

It uncovers a singularity in the large deviation function indicating a phase transition and relates short-time tail behaviors to long-time distributions in the KPZ context.

Findings

Identifies a second-order phase transition at a critical height $H_c$.

Shows tail scaling laws for large deviations match known distributions.

Connects short-time behavior with long-time distribution tails.

Abstract

We study the short-time behavior of the probability distribution $\mathcal{P}(H,t)$ of the surface height $h(x=0,t)=H$ in the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimension. The process starts from a stationary interface: $h(x,t=0)$ is given by a realization of two-sided Brownian motion constrained by $h(0,0)=0$. We find a singularity of the large deviation function of $H$ at a critical value $H=H_c$. The singularity has the character of a second-order phase transition. It reflects spontaneous breaking of the reflection symmetry $x \leftrightarrow -x$ of optimal paths $h(x,t)$ predicted by the weak-noise theory of the KPZ equation. At $|H|\gg |H_c|$ the corresponding tail of $\mathcal{P}(H)$ scales as $-\ln \mathcal{P} \sim |H|^{3/2}/t^{1/2}$ and agrees, at any $t>0$, with the proper tail of the Baik-Rains distribution, previously observed only at long times. The other tail of $\mathcal{P}$ scales as $-\ln \mathcal{P} \sim |H|^{5/2}/t^{1/2}$ and coincides with the corresponding tail for the sharp-wedge initial condition.