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

Abstract

This review is devoted to detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different types of the free energy probability distribution functions. In the second part of the review we discuss the problems which are still waiting for solutions.

Velocity distribution functions and intermittency in one-dimensional randomly forced Burgers turbulence

“…(A.13) We obtain the rest of the derivatives in a similar way: We see that in the limit → 0, according to the above expressions-equations (A.13)-(A.26)-the only non-zero contributions to the function P x (v, v ) in equations (101)-( 102) are given by the two terms G 123 ∝ 1/ 2 , equation (A. 19), and the product G 3 S 12 ∝ 1/ 2 , equations (A.15) and (A.23).…”

“…The high temperature limit of the replica problem formulated above is well studied (for a review see e.g. [19] and references therein). It can be shown that in the limit β → 0 the interaction potential U (x), eq.…”

“…The high temperature limit of the replica problem formulated above is well studied (for a review see e.g. [19] and references therein). It can be shown that in the limit β → 0, the interaction potential U (x), equation (6), can be approximated by the δ-function, and in this case the generic solution of the Schrödinger equation can be represented in terms of the Bethe ansatz eigenfunctions [20][21][22].…”

Perturbative solution for the spectral gap of the weakly asymmetric exclusion process

“…Under the previous assumptions, the Euler-Maclaurin calculations leading to the edge Bethe equations (14), (13) can presumably be made rigorous without too much effort. Additionally, the limit µ → 0 of the edge Bethe equations (15) uses the unproven assumption that w 0 (µ) goes to infinity when µ → 0, while the w j (µ), j ∈ Z * converge to distinct finite values w j (0).…”

“…ASEP belongs to the prominent non-equilibrium universality class known as KPZ [6,7,8,9,10,11,12,13,14], from Kardar, Parisi and Zhang [15], and which has been known to describes specific regimes of growing interfaces, driven lattice gases and directed polymers in random media. More recently, KPZ universality has been extended to several other settings featuring a strong interplay between noise and non-linearity, in particular one-dimensional systems with few conservation laws in the framework of nonlinear fluctuating hydrodynamics [16,17,18] (including classical fluids [19], anharmonic chains [20,21,22] and quantum liquids described by the Gross-Pitaevskii equation [23,24]), as well as two-dimensional strongly localized systems [25,26] and free Fermions in a harmonic trap [27,28].…”