Spatial correlations of the 1D KPZ surface on a flat substrate

TL;DR

This paper analyzes the spatial correlations of the 1D KPZ surface with flat initial conditions, deriving explicit formulas for multi-point distributions and connecting to eigenvalue dynamics in random matrix theory.

Contribution

It provides an explicit Fredholm determinant representation for the multi-point height distribution of the KPZ surface, linking it to GOE Dyson's Brownian motion.

Findings

Derived the multi-point joint distribution as a Fredholm determinant.

Connected KPZ surface correlations to GOE Dyson's Brownian motion.

Reformulated Green's function for ASEP via vicious walk problem.

Abstract

We study the spatial correlations of the one-dimensional KPZ surface for the flat initial condition. It is shown that the multi-point joint distribution for the height is given by a Fredholm determinant, with its kernel in the scaling limit explicitly obtained. This may also describe the dynamics of the largest eigenvalue in the GOE Dyson's Brownian motion model. Our analysis is based on a reformulation of the determinantal Green's function for the totally ASEP in terms of a vicious walk problem.