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

TL;DR

The paper investigates the extreme statistics of N independent random walks in a space-time random environment and reveals a super-universal KPZ-like structure governed by the lowest random moment of the environment. By mapping tail and extreme statistics to the stochastic heat equation and KPZ via a replica/tilting framework, it derives how means depend on the diffusion coefficient while variances split into environment-driven and sampling components. The key finding is that the leading-scale behavior of extreme location and extreme first passage time is universal within universality classes labeled by m, with explicit formulas for the mean and variance that involve an extreme-diffusion coefficient $D_{ext}$ and the noise strength $D_0$. The results show that higher environmental moments imprint anomalous scaling and that the onset of KPZ fluctuations persists across a wide range of environmental models, providing a method to infer fine-grained environment fluctuations from extreme-diffusion data. Numerics across diverse environmental constructions corroborate the theoretical predictions, highlighting practical implications for diagnosing microstructure in diffusive systems.

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.

Figures (2)

• Figure 1: Several examples of random environments. Blue boxes are lattice sites and the opacity of the gray arrows shows the probability of jumping to a site. Large, solid arrows show the average drift at each site. The size of the arrow scales with the variance of the jump distribution and the color denotes the skew (cyan skewed down and pink skewed up). (a) Environment where the drift, variance and skewness are random. (b) Environment with no drift, but the variance and skewness are random. (c) Environment with no drift, a constant variance but a random skewness.
• Figure 2: Plot of the different variances for the extreme location (a) and the extreme first passage time (d), for the symmetric Dirichlet distribution and systems of size $N=10^{28}$. Dashed lines represent my theoretical predictions and solid lines are numerically measured values. Collapse of the environmental fluctuations of the extreme location (b) and extreme first passage time (e) for $m=1$ (red), $m=2$ (green), $m=3$ (purple) and $m=4$ (blue). I plot a number of different distributions listed in the "End Matter". The saturation of each curve is scaled by the diffusion coefficient, $D$. The ratio of measured $\mathrm{Env}_t^N$ (c) and $\mathrm{Env}_L^N$ (f) to their theoretical predictions in Eq. \ref{['eq:envMax']} and \ref{['eq:envFPT']}, respectively. All curves are computed over 500 independent environments.