# Diffusion in time-dependent random media and the Kardar-Parisi-Zhang equation

## TL;DR

This paper explores how diffusion in time-dependent random media relates to the KPZ universality class, predicting universal fluctuation distributions across dimensions and verifying these predictions numerically.

## Contribution

It provides a continuum and lattice model framework linking diffusion in random media to KPZ universality, including predictions of universal distributions in various dimensions.

## Key findings

- GOE Tracy-Widom distribution for 1D transition probability fluctuations
- Phase transition from Gaussian to KPZ fluctuations in 3D with increasing bias
- KPZ universal distributions for first particle arrival times

## Abstract

Although time-dependent random media with short range correlations lead to (possibly biased) normal tracer diffusion, anomalous fluctuations occur away from the most probable direction. This was pointed out recently in 1D lattice random walks, where statistics related to the 1D Kardar- Parisi-Zhang (KPZ) universality class, i.e. the GUE Tracy Widom distribution, were shown to arise. Here we provide a simple picture for this correspondence, directly in the continuum as well as for lattice models, which allows to study arbitrary space dimension and to predict a variety of universal distributions. In $d = 1$ we predict and verify numerically the emergence of the GOE Tracy-Widom distribution for the fluctuations of the transition probability. In $d = 3$ we predict a phase transition from Gaussian fluctuations to 3D-KPZ type fluctuations as the bias is increased. We predict KPZ universal distributions for the arrival time of a first particle from a cloud diffusing in such media.