The probability density function tail of the Kardar-Parisi-Zhang equation in the strongly non-linear regime

TL;DR

This paper analytically derives the tail of the probability density function for interface growth governed by the KPZ equation, revealing a Tracy-Widom distribution and discussing the impact of spatial dimensions.

Contribution

It provides a novel analytical derivation of the PDF tail for the KPZ equation using the instanton method, connecting it to Tracy-Widom distribution and exploring dimensional effects.

Findings

PDF tail matches Tracy-Widom distribution proportional to exp(-c w_2^{3/2})

No upper critical dimension for the KPZ interface growth

Method offers new insights into the rightmost tail of the interface width distribution

Abstract

An analytical derivation of the probability density function (PDF) tail describing the strongly correlated interface growth governed by the nonlinear Kardar-Parisi-Zhang equation is provided. The PDF tail exactly coincides with a Tracy-Widom distribution i.e. a PDF tail proportional to $\exp( - c w_2^{3/2})$, where $w_2$ is the the width of the interface. The PDF tail is computed by the instanton method in the strongly non-linear regime within the Martin-Siggia-Rose framework using a careful treatment of the non-linear interactions. In addition, the effect of spatial dimensions on the PDF tail scaling is discussed. This gives a novel approach to understand the rightmost PDF tail of the interface width distribution and the analysis suggests that there is no upper critical dimension.