Exact scaling functions for one-dimensional stationary KPZ growth

TL;DR

This paper derives exact scaling functions for the stationary two-point correlation in one-dimensional KPZ growth by connecting a microscopic model to a Riemann-Hilbert problem involving Painleve II, providing precise numerical solutions.

Contribution

It introduces an exact method to compute the stationary correlation function of 1D KPZ growth using integrable systems and Riemann-Hilbert analysis, surpassing previous approximation methods.

Findings

Exact scaling function computed with high precision.

Numerical results agree with mode coupling predictions.

Provides a new analytical approach to KPZ correlations.

Abstract

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painleve II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation.