Stochastic growth in time dependent environments

TL;DR

This paper investigates how time-dependent noise variance affects the KPZ growth equation in one dimension, revealing a phase transition at a critical decay rate of noise variance with distinct universal and non-universal behaviors.

Contribution

It introduces a comprehensive analysis of KPZ with time-dependent noise variance, identifying a phase transition and employing multiple methods including exact solutions and simulations.

Findings

Transition at a=1/2 in noise decay rate

Saturation to non-universal distribution for a>1/2

Scaling exponents depend on a for a<1/2

Abstract

We study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance $c(t)$ depending on time. We find that for $c(t)\propto t^{-\alpha}$ there is a transition at $\alpha=1/2$. When $\alpha>1/2$, the solution saturates at large times towards a non-universal limiting distribution. When $\alpha<1/2$ the fluctuation field is governed by scaling exponents depending on $\alpha$ and the limiting statistics are similar to the case when $c(t)$ is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.