Probability distribution of the free energy of a directed polymer in a random medium

Abstract

We calculate exactly the first cumulants of the free energy of a directed polymer in a random medium for the geometry of a cylinder. By using the fact that the n-th moment <Z^n> of the partition function is given by the ground state energy of a quantum problem of n interacting particles on a ring of length L, we write an integral equation allowing to expand these moments in powers of the strength of the disorder gamma or in powers of n. For n small and n of order (L gamma)^(-1/2), the moments <Z^n> take a scaling form which allows to describe all the fluctuations of order 1/L of the free energy per unit length of the directed polymer. The distribution of these fluctuations is the same as the one found recently in the asymmetric exclusion process, indicating that it is characteristic of all the systems described by the Kardar-Parisi-Zhang equation in 1+1 dimensions.

18 pages, no figure, tu appear in PRE 61 (2000)

References in corpus (2)

• Statistical Self-Similarity of One-Dimensional Growth Processes
• Ground state energy of a non-integer number of particles with delta attractive interactions

Cited by in corpus (67)

• Free-energy distribution of the directed polymer at high temperature
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• An exact solution for the KPZ equation with flat initial conditions
• Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
• Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature
• The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
• Majorizing multiplicative cascades for directed polymers in random media
• Explicit solution of the optimal fluctuation problem for an elastic string in random potential
• Distribution of the time at which N vicious walkers reach their maximal height
• Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers
• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Finite-time fluctuations for the totally asymmetric exclusion process
• The Asymmetric Avalanche Process
• Current Fluctuations in the exclusion process and Bethe Ansatz
• Two-point free energy distribution function in (1+1) directed polymers
• Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets
• On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window
• Large deviations in spin-glass ground-state energies
• Bethe ansatz solution of discrete time stochastic processes with fully parallel update
• Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation
• Two-time free energy distribution function in (1+1) directed polymers
• Attempted Bethe ansatz solution for one-dimensional directed polymers in random media
• Two-point generating function of the free energy for a directed polymer in a random medium
• Derivation of the Functional Renormalization Group Beta-Function at order 1/N for Manifolds Pinned by Disorder
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Tail of the two-time height distribution for KPZ growth in one dimension
• Probability Distributions of Line Lattices in Random Media from the 1D Bose Gas
• The universal high temperature regime of pinned elastic objects
• Riemann surfaces for KPZ with periodic boundaries
• Intermittency of Height Fluctuations and Velocity Increment of The Kardar-Parisi-Zhang and Burgers Equations with infinitesimal surface tension and Viscosity in 1+1 Dimensions
• Statistical Theory for the Kardar-Parisi-Zhang Equation in 1+1 Dimension
• Replica symmetry breaking in trajectories of a driven Brownian particle
• Dynamic criticality far-from-equilibrium: one-loop flow of Burgers-Kardar-Parisi-Zhang systems with broken Galilean invariance
• The crossing probability for directed polymers in random media
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice - bulk eigenvalues
• Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)
• Fluctuations of the winding number of a directed polymer in a random medium
• Test of Replica Theory: Thermodynamics of 2D Model Systems with Quenched Disorder
• Field theory of disordered systems -- Avalanches of an elastic interface in a random medium
• Integrable stochastic dualities and the deformed Knizhnik-Zamolodchikov equation
• Extrapolation methods and Bethe ansatz for the asymmetric exclusion process
• On two-time distribution functions in (1+1) random directed polymers
• $N$-point free energy distribution function in one dimensional random directed polymers
• Current fluctuations and large deviations for periodic TASEP on the relaxation scale
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Riemann surface for TASEP with periodic boundaries
• Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential
• Asymptotics for the norm of Bethe eigenstates in the periodic totally asymmetric exclusion process
• Directed polymers and interfaces in random media : free-energy optimization via confinement in a wandering tube
• High temperature behaviors of the directed polymer on a cylinder
• KPZ fluctuations in finite volume
• Correlation Functions for an Elastic String in a Random Potential: Instanton Approach
• Log-gamma directed polymer with one free end via coordinate Bethe Ansatz
• Distribution of Brownian coincidences
• Current statistics in the q-boson zero range process
• Analytical Methods and Field Theory for Disordered Systems
• KPZ on torus: Gaussian fluctuations
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• Numerical Method for Accessing the Universal Scaling Function for a Multi-Particle Discrete Time Asymmetric Exclusion Process
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Two-time distribution functions in the Gaussian model of randomly forced Burgers turbulence
• Absence of weak disorder for directed polymers on supercritical percolation clusters
• Central limit theorems for the (2+1)-dimensional directed polymer in the weak disorder limit
• Two-temperature statistics of free energies in (1+1) directed polymers
• Crossover scaling functions in the asymmetric avalanche process
• Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz