On Time Correlations for KPZ Growth in One Dimension

TL;DR

This paper analyzes time correlations in 1+1 dimensional KPZ growth, deriving power laws for covariance under various initial conditions using Airy processes and variational methods.

Contribution

It provides new exact formulas and power law behaviors for time covariances in KPZ growth, including stationary initial conditions, via variational and lattice gas approaches.

Findings

Power laws for short and long time covariances are derived.

Exact stationary covariance formulas are obtained.

Large time behavior of height gradients is characterized.

Abstract

Time correlations for KPZ growth in 1+1 dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the l arge time behavior for the covariance of the height gradients.