Super-Universal Behavior of Outliers Diffusing in a Space-Time Random Environment

TL;DR

This paper investigates the extreme behavior of particles diffusing in a space-time random environment, revealing universal scaling laws governed by the KPZ equation and demonstrating a super-universality class across different environmental moments.

Contribution

It introduces a comprehensive framework linking extreme diffusion statistics to the KPZ universality class and characterizes how environmental moments influence scaling regimes.

Findings

Extreme location and first passage time follow KPZ statistics.

Scaling regimes depend on the environment's moments.

Numerical simulations confirm theoretical predictions.

Abstract

I characterize the extreme location and extreme first passage time of a system of $N$ particles independently diffusing in a space-time random environment. I show these extreme statistics are governed by the Kardar-Parisi-Zhang (KPZ) equation and derive their mean and variance. I find the scalings of the statistics depend on the moments of the environment. Each scaling regime forms a universality class which is controlled by the lowest order moment which exhibits random fluctuations. When the first moment is random, the environment plays the role of a random velocity field. When the first moment is fixed but the second moment is random, the environment manifests as fluctuations in the diffusion coefficient. As each higher moment is fixed, the next moment determines the scaling behavior. Since each scaling regime forms a universality class, this model for diffusion forms a super-universality class. I confirm my theoretical predictions using numerics for a wide class of underlying environments.