Directed polymer near a hard wall and KPZ equation in the half-space

CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure - 24 rue Lhomond, 75231 Cedex 05, Paris, France, EU

Received: 10 September 2012
Accepted: 5 October 2012

Abstract

We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in the presence of disorder, equivalent to the KPZ equation on the half-space with droplet initial conditions. From a Bethe Ansatz solution of the equivalent attractive boson model we obtain the exact expression for the free-energy distribution at all times. It converges at large time to the Tracy-Widom distribution F₄ of the Gaussian symplectic ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.