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

Understand

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.