Two-point generating function of the free energy for a directed polymer in a random medium

We consider a (1 + 1)-dimensional directed continuum polymer in a Gaussian delta-correlated spacetime random potential. For this model the moments (= replica) of the partition function, Z(x,t), can be expressed in terms of the attractive δ-Bose gas on the line. Based on a recent study of the structure of the eigenfunctions, we compute the generating function for Z(x ₁,t), Z(x₂,t) under a particular decoupling assumption and thereby extend recent results on the one-point generating function of the free energy to two points. It is established that in the long-time limit the fluctuations of the free energy are governed by the two-point distribution of the Airy process, which further supports that the longtime behavior of the KPZ equation is the same as derived previously for lattice growth models.