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

TL;DR

This paper derives exact results for the free energy distribution of a directed polymer near a hard wall, showing convergence to the Tracy-Widom GSE distribution, and compares these with numerical simulations.

Contribution

It provides an exact solution for the free energy distribution of the directed polymer in a half-space using Bethe Ansatz, connecting it to the KPZ universality class.

Findings

Free energy distribution converges to Tracy-Widom GSE at large times.

Exact expression for the free energy distribution at all times.

Numerical simulations confirm theoretical predictions.

Abstract

We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in 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 $F_4$ of the Gaussian Symplectic Ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.