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.

45 pages, 3 figures

References in corpus (12)

• Exact scaling functions for one-dimensional stationary KPZ growth
• An exact solution for the KPZ equation with flat initial conditions
• Full counting statistics in a propagating quantum front and random matrix spectra
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Exact distribution of the maximal height of p vicious walkers
• Nonintersecting Brownian excursions
• The One-dimensional KPZ Equation and the Airy Process
• Memory and universality in interface growth
• Maximum distributions of bridges of noncolliding Brownian paths
• Two-point generating function of the free energy for a directed polymer in a random medium

Cited by in corpus (2)

• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• KPZ fluctuations in finite volume