Exact height distributions for the KPZ equation with narrow wedge initial condition

TL;DR

This paper derives an exact formula for the height distribution of the KPZ equation with narrow wedge initial condition, showing convergence to Tracy-Widom distribution and providing numerical validation.

Contribution

It presents a determinantal formula for the one-point distribution of the KPZ height with narrow wedge initial condition, connecting it to integrable probability results.

Findings

Distribution converges to Tracy-Widom distribution as time goes to infinity.

Provides an exact formula valid for any position and time.

Numerical computations confirm theoretical predictions.

Abstract

We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution $h(x,t)$ valid for any $x$ and $t>0$. The corresponding distribution function converges in the long time limit, $t\to\infty$, to the Tracy-Widom distribution. The first order correction is a shift of order $t^{-1/3}$. We provide numerical computations based on the exact formula.