Tail of the two-time height distribution for KPZ growth in one dimension

Abstract

Obtaining the exact multi-time correlations for one-dimensional growth models described by the Kardar-Parisi-Zhang (KPZ) universality class is presently an outstanding open problem. Here, we study the joint probability distribution function (JPDF) of the height of the KPZ equation with droplet initial conditions, at two different times $t_1<t_2$, in the limit where both times are large and their ratio $t_2/t_1$ is fixed. This maps to the JPDF of the free energies of two directed polymers with two different lengths and in the same random potential. Using the replica Bethe ansatz (RBA) method, we obtain the exact tail of the JPDF when one of its argument (the KPZ height at the earlier time $t_1$) is large and positive. Our formula interpolates between two limits where the JPDF decouples: (i) for $t_2/t_1 \to +\infty$ into a product of two GUE Tracy-Widom (TW) distributions, and (ii) for $t_2/t_1 \to 1^+$ into a product of a GUE-TW distribution and a Baik-Rains distribution (associated to stationary KPZ evolution). The lowest cumulants of the height at time $t_2$, conditioned on the one at time $t_1$, are expressed analytically as expansions around these limits, and computed numerically for arbitrary $t_2/t_1$. Moreover we compute the connected two-time correlation, conditioned to a large enough value at $t_1$, providing a quantitative prediction for the so-called persistence of correlations (or ergodicity breaking) in the time evolution from the droplet initial condition. Our RBA results are then compared with arguments based on Airy processes, with satisfactory agreement. These predictions are universal for all models in the KPZ class and should be testable in experiments and numerical simulations.