N-point free energy distribution function in one dimensional random directed polymers

Dotsenko Dotsenko

doi:10.5488/cmp.17.33003

Abstract

Explicit expression for the $N$-point free energy distribution function in one dimensional directed polymers in a random potential is derived in terms of the Bethe ansatz replica technique. The obtained result is equivalent to the one derived earlier by Prolhac and Spohn [J. Stat. Mech., 2011, P03020].

Highlights

In this paper we consider the model of one-dimensional directed polymers in a quenched random potential. This model is defined in terms of an elastic string φ(τ) directed along the τ-axis within an interval [0, t ] which passes through a random medium described by a random potential V (φ, τ)
In the limit t → ∞, the polymer mean squared displacement exhibits a universal scaling form 〈φ2〉 ∝ t 4/3 (where 〈. . . 〉 and (. . . ) denote the thermal and the disorder averages) while the typical value of the free energy fluctuations scales as t 1/3
It is the statistics of rescaled free energy fluctuations f (x) which in the limit t → ∞ is expected to be described by a non-trivial universal distribution W ( f )

Summary

Introduction

In this paper we consider the model of one-dimensional directed polymers in a quenched random potential. Where the disorder potential V [φ, τ] is described by the Gaussian distribution with a zero mean V (φ, τ) = 0 and the δ-correlations: V (φ, τ)V (φ′, τ′) = uδ(τ − τ′)δ(φ − φ′) The parameter u describes the strength of the disorder. The system of this type as well as the equivalent problem of the KPZ-equation [1] describing the growth of an interface with time in the presence of noise have been the subject of intense investigations for about the last three decades Recently the two-point free energy distribution function which describes the joint statistics of the free energies of the directed polymers coming to two different endpoints has been derived in [29,30,31]

Dotsenko

N -point distribution function

Conclusions