Extrema statistics in the dynamics of a non-Gaussian random field

TL;DR

This paper introduces a framework to analyze how nonlinear dynamics induce non-Gaussian features in the statistical distribution of a evolving random field, focusing on maxima and minima densities.

Contribution

It provides a geometrical method to quantify non-Gaussian effects and nonlinear terms in the evolution of random fields, demonstrated on the KPZ surface growth model.

Findings

Framework effectively detects non-Gaussian contributions.

Application to KPZ shows nonlinear dynamics alter maxima/minima densities.

Method can quantify the magnitude of nonlinear effects.

Abstract

When the equations that govern the dynamics of a random field are nonlinear, the field can develop with time non-Gaussian statistics even if its initial condition is Gaussian. Here, we provide a general framework for calculating the effect of the underlying nonlinear dynamics on the relative densities of maxima and minima of the field. Using this simple geometrical probe, we can identify the size of the non-Gaussian contributions in the random field, or alternatively the magnitude of the nonlinear terms in the underlying equations of motion. We demonstrate our approach by applying it to an initially Gaussian field that evolves according to the deterministic KPZ equation, which models surface growth and shock dynamics.