Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 074, 23 pages      arXiv:1602.00486      https://doi.org/10.3842/SIGMA.2016.074
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

On Time Correlations for KPZ Growth in One Dimension

Patrik L. Ferrari a and Herbert Spohn b
a) Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany
b) Zentrum Mathematik, TU München, Boltzmannstrasse 3, D-85747 Garching, Germany

Received March 17, 2016, in final form July 21, 2016; Published online July 26, 2016

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 large time behavior for the covariance of the height gradients.

Key words: KPZ universality, space-time correlations, interacting particles, last passage percolation.

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