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

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.