How flat is flat in random interface growth?

TL;DR

This paper characterizes the universality classes of the KPZ equation's one-point fluctuations based on initial data, distinguishing flat, curved, and Brownian types through a large deviation criterion and providing explicit formulas for their distributions.

Contribution

It introduces a large deviation criterion to classify initial data into universality classes for KPZ fluctuations and extends the analysis of the Airy$_2$ process to general initial conditions.

Findings

Identifies domains of attraction for flat, curved, and Brownian initial data.

Provides a variational formula for the distribution of fluctuations.

Derives an explicit Fredholm determinant formula for square root initial data.

Abstract

Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on a large deviation rate function for the rescaled initial data, which arises naturally from the Hopf-Cole transformation. This allows us, in particular, to distinguish the domains of attraction of curved, flat, and Brownian initial data, and to identify the boundary between the curved and flat domains of attraction, which turns out to correspond to square root initial data. The distribution of the asymptotic one-point fluctuations is characterized by means of a variational formula written in terms of certain limiting processes (arising as subsequential limits of the spatial fluctuations of KPZ equation with narrow wedge initial data, as shown in [CH16]) which are widely believed to coincide with the Airy$_2$ process. In order to identify these distributions for general initial data, we extend earlier results on continuum statistics of the Airy$_2$ process to probabilities involving the process on the entire line. In particular, this allows us to write an explicit Fredholm determinant formula for the case of square root initial data.