Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz

We show how our previous result based on the replica Bethe ansatz for the Kardar Parisi Zhang (KPZ) equation with the “half-flat” initial condition leads to the Airy${}_2$ to Airy${}_1$ (i.e. GUE to GOE) universal crossover one-point height distribution in the limit of large time. Equivalently, we obtain the distribution of the free energy of a long directed polymer (DP) in a random potential with one fixed endpoint and the other one on a half-line. We then generalize to a DP when each endpoint is free on its own half-line. It amounts, in the limit of large time, to obtain the distribution of the maximum of the transition process Airy${}_{2\to 1}$ (minus a half-parabola) on a half line.