Exact free energy distribution function of a randomly forced directed polymer

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 $\mathcal{P}[F(u,L)]$ for the free energy $F$ of a string starting at $(0,0)$ and ending at $(u,L)$. The function $\mathcal{P}(F)$ is strongly asymmetric, with the left tail decaying exponentially ($\ln \mathcal{P}(F\to -\infty )\propto F$) and the right tail vanishing as $\ln \mathcal{P}(F\to +\infty )\propto -F^3$. Our analysis defines a strategy for future attacks on this class of problems.

RevTeX, 4 pages, 1 figure inserted

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