Free-energy distribution of the directed polymer at high temperature

TL;DR

This paper analyzes the free-energy distribution of a directed polymer in a random potential at high temperature, deriving exact results and showing a crossover to Tracy-Widom distribution at large times.

Contribution

It provides an exact expression for the free energy distribution at any time and demonstrates the crossover to Tracy-Widom distribution in the high-temperature regime.

Findings

Exact moments of the partition function obtained via Bethe Ansatz.

Free energy distribution crosses over to Tracy-Widom distribution at large times.

Agreement between analytical predictions and numerical simulations.

Abstract

We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.