Exact free energy distribution function of a randomly forced directed polymer

We study the elastic (1 + 1)-dimensional string subject to a random Gaussia n potential on scales smaller than the correlation radius of the disorder p otential (Larkin problem). We present an exact calculation of the probabili ty function P([F[u, L]) for the free energy F of a string starting at (0, 0 ) and ending at (u, L). The function P(F) is strongly asymmetric, with the left tail decaying exponentially [In P(F --> -infinity) proportional to F] and the right tail vanishing as In P(F --> +infinity) proportional to -F-3. Our analysis defines a strategy for future attacks on this class of proble ms. [S0031-9007(99)08838-9].