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

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.

19 pages, 8 figures

References in corpus (7)

• Free-energy distribution of the directed polymer at high temperature
• Asymptotics in ASEP with Step Initial Condition
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• The crossover regime for the weakly asymmetric simple exclusion process
• Total Current Fluctuations in ASEP
• Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers
• Formulas for ASEP with Two-Sided Bernoulli Initial Condition

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