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

Abstract

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 Airy2 to Airy1 (i.e. GUE (Gaussian unitary ensemble) to GOE (Gaussian orthogonal ensemble)) universal crossover one-point height distribution in the limit of large time. It involves a ‘decoupling assumption’ in that limit, validated by the result. 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 generalize to a DP when each endpoint is free on its own half-line. This yields, in the large time limit, a conjecture for the distribution of the maximum of the transition process Airy2→1 (minus a half-parabola) on a half-line.