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 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 (ln P(F → −∞) ∝ F ) and the right tail vanishing as ln P(F → +∞) ∝ −F 3 . Our analysis defines a strategy for future attacks on this class of problems.

One more discussion of the replica trick: the example of the exact solution

“…Thus, according to Equation (55), for the distribution function of the rescaled free energy fluctuations of the infinite system, P Ã ð f Þ lim L!1 P L ð f Þ, we obtain the following (universal) result [33]:…”

“…The thermodynamic limit, L ! 1, of this system has been studied earlier [33]. Here I would like to concentrate on the technical details of the calculations.…”

Statistical Properties of One-Dimensional Directed Polymers in a Random Potential

“…In the present model this distribution can be computed explicitly and the resulting function P x|y;t (F ) turns out to be rather non-trivial [76,116,117]. Mapping the considered system to quantum bosons (see Chapter III) one introduces the N -particle wave function:…”

Statistical theory for the Kardar-Parisi-Zhang equation in(1+1)dimensions

“…It has been shown in [34] that, one can map the kinetics of the annihilation process A + B → 0 with driven diffusion onto the (1+1)-dimensional KPZ equation. Also the KPZ equation is closely related to the dynamics of a sine-Gordon chain [35], the driven-diffusion equation [36][37], high T csuperconductor [38] and directed paths in the random media [39][40][41][42][43][44][45][46][47][48][49][50][51][52] and charge density waves [53], dislocations in disordered solids [3], formation of large-scale structure in the universe [54][55][56][57] , Burgers turbulence 90] and etc.…”