  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.
