Maximum of an Airy process plus Brownian motion and memory in KPZ growth

TL;DR

This paper derives exact universal distribution results for the maximum of the Airy$_2$ process plus Brownian motion, providing insights into memory effects and correlations in the KPZ growth universality class through replica Bethe ansatz methods.

Contribution

It introduces new exact results for distributions involving the Airy process and Brownian motion in KPZ, including two-time correlations and midpoint distributions, using a novel decoupling assumption.

Findings

Universal limit of two-time height correlation is approximately 0.623.

Distribution of the midpoint position of directed polymers is characterized.

Height distribution in stationary KPZ with a step is obtained.

Abstract

We obtain several exact results for universal distributions involving the maximum of the Airy$_2$ process minus a parabola and plus a Brownian motion, with applications to the 1D Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g. $C_{\infty} = \lim_{t_2/t_1 \to +\infty} \overline{h(t_1) h(t_2)}^c/\overline{h(t_1)^2}^c$, with $C_{\infty} \approx 0.623$, as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position $x(t_1)$ of a directed polymer of length $t_2$, and (iii) the height distribution in stationary KPZ with a step. These results are derived from the replica Bethe ansatz for the KPZ continuum equation, with a "decoupling assumption" in the large time limit. They agree and confirm, whenever they can be compared, with (i) our recent tail results for two-time KPZ with de Nardis, checked in experiments with Takeuchi, (ii) a recent result of Maes and Thiery on midpoint position.