The height distribution of the KPZ equation with sharp wedge initial condition: numerical evaluations

TL;DR

This paper numerically evaluates the height distribution of the KPZ equation with sharp wedge initial conditions, illustrating the transition from Gaussian to GUE Tracy-Widom distributions over time.

Contribution

It provides a numerical analysis of the KPZ height distribution evolution, confirming theoretical predictions of the crossover behavior.

Findings

Clear visualization of the Gaussian to Tracy-Widom transition

Numerical validation of the convolution formula for the distribution

Demonstration of the distribution's behavior over the entire time span

Abstract

The time-dependent probability distribution function of the height for the Kardar-Parisi-Zhang equation with sharp wedge initial conditions has been obtained recently as a convolution between the Gumbel distribution and a difference of two Fredholm determinants. We evaluate numerically this distribution over the whole time span. The crossover from the short time behavior, which is Gaussian, to the long time behavior, which is governed by the GUE Tracy-Widom distribution, is clearly visible.