Exact free energy distribution function of a randomly forced directed polymer

TL;DR

This paper derives an exact probability distribution function for the free energy of a (1+1)-dimensional directed polymer in a Gaussian random potential, revealing asymmetric tails and providing a new analytical approach.

Contribution

It presents the first exact calculation of the free energy distribution for a directed polymer in a Gaussian disorder, including tail behaviors.

Findings

Left tail decays exponentially with F

Right tail vanishes as exp(-F^3)

Distribution is strongly asymmetric

Abstract

We study the elastic (1+1)-dimensional string subject to a random gaussian potential on scales smaller than the correlation radius of the disorder potential (Larkin problem). We present an exact calculation of the probability function ${\cal P} [F(u,L)]$ for the free energy $F$ of a string starting at $(0,0)$ and ending at $(u,L)$. The function ${\cal P}(F)$ is strongly asymmetric, with the left tail decaying exponentially ($\ln {\cal P}(F\to-\infty)\propto F$) and the right tail vanishing as $\ln {\cal P}(F\to +\infty)\propto -F^{3}$. Our analysis defines a strategy for future attacks on this class of problems.