Abstract
We obtain several exact results for universal distributions involving the maximum of the process minus a parabola and plus a Brownian motion, with applications to the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic growth universality class. This allows one to obtain (i) the universal limit, for large time separation, of the two-time height correlation for droplet initial conditions, e.g., , with , as well as conditional moments, which quantify ergodicity breaking in the time evolution; (ii) in the same limit, the distribution of the midpoint position of a directed polymer of length ; 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 the work by de Nardis and Le Doussal [J. Stat. Mech. (2017) 053212], checked in experiments with the work by Takeuchi and co-workers [De Nardis et al., Phys. Rev. Lett. 118, 125701 (2017)] and (ii) a recent result of Maes and Thiery [J. Stat. Phys. 168, 937 (2017)] on midpoint position.
- Received 28 September 2017
DOI:https://doi.org/10.1103/PhysRevE.96.060101
©2017 American Physical Society