Time-time covariance for last passage percolation with generic initial profile

TL;DR

This paper rigorously analyzes the time correlation in KPZ growth models near characteristics, proving covariance convergence for various initial profiles and establishing universality of correction terms, extending results to non-stationary profiles.

Contribution

It provides a rigorous proof of covariance formulas for KPZ growth with different initial profiles and demonstrates universality of correction terms for close observation times.

Findings

Proves covariance convergence for droplet, flat, and stationary initial profiles.

Establishes universality of first order correction for close observation times.

Extends covariance results to non-stationary initial profiles.

Abstract

We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence of the covariance with droplet, flat and stationary initial profile. In particular, this provides a rigorous proof of the exact formula of the covariance for the stationary case obtained in [SIGMA 12 (2016), 074]. Furthermore, we prove the universality of the first order correction when the two observation times are close and provide a rigorous bound of the error term. This result holds also for random initial profiles which are not necessarily stationary.