The KPZ equation with flat initial condition and the directed polymer with one freeend

Abstract

We study the directed polymer (DP) of lengtht in a random potentialin dimension 1 + 1 in the continuum limit, with one end fixed and one end free. Thismaps onto the Kardar–Parisi–Zhang growth equation in timet, with flat initial conditions. We use the Bethe ansatz solution for the replicated problem,which is an attractive bosonic model. The problem is more difficult than the previoussolution of the fixed endpoint problem as it requires regularization of the spatialintegrals over the Bethe eigenfunctions. We use either a large fixed system length orsmall finite slope KPZ initial conditions (wedge). The latter allows one to takeproperly into account non-trivial contributions, which appear as deformed strings inthe former. By considering a half-space model in a proper limit we obtain anexpression for the generating function of all positive integer moments of the directed polymer partition function. We obtain the generating function of the moments of theDP partition sum as a Fredholm Pfaffian. At large time, this Fredholm Pfaffian, valid for all timet, exhibits convergence of the free energy (i.e. KPZ height) distribution to the GOETracy–Widom distribution.