Persistence of Kardar-Parisi-Zhang Interfaces

TL;DR

This paper investigates the persistence probabilities of KPZ interfaces, revealing decay behaviors, deriving bounds, and exploring oscillatory phenomena, thus advancing understanding of interface dynamics in non-equilibrium growth models.

Contribution

It provides numerical estimates of persistence exponents, derives bounds from autocorrelation functions, and uncovers oscillatory persistence in a discretized KPZ model.

Findings

Persistence probabilities decay as power laws with specific exponents.

Bounds on exponents are derived from height autocorrelation functions.

Oscillatory persistence probabilities indicate hidden temporal correlations.

Abstract

The probabilities $P_\pm(t_0,t)$ that a growing Kardar-Parisi-Zhang interface remains above or below the mean height in the time interval $(t_0, t)$ are shown numerically to decay as $P_\pm \sim (t_0/t)^{\theta_\pm}$ with $\theta_+ = 1.18 \pm 0.08$ and $\theta_- = 1.64 \pm 0.08$. Bounds on $\theta_\pm$ are derived from the height autocorrelation function under the assumption of Gaussian statistics. The autocorrelation exponent $\bar \lambda$ for a $d$--dimensional interface with roughness and dynamic exponents $\beta$ and $z$ is conjectured to be $\bar \lambda = \beta + d/z$. For a recently proposed discretization of the KPZ equation we find oscillatory persistence probabilities, indicating hidden temporal correlations.