Free-energy distribution of the directed polymer at high temperature

Abstract

We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature $T$ is described, upon scaling 'time' $t\sim T^5/\kappa$ and space $x=T^3/\kappa$ (with $\kappa =T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.

6 pages, 3 figures large time limit corrected and convergence to Tracy Widom established, 1 figure changed.

References in corpus (22)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Burgers Turbulence
• Exact scaling functions for one-dimensional stationary KPZ growth
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• Correlation functions of the one-dimensional attractive Bose gas
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Probability distribution of the free energy of a directed polymer in a random medium
• Equilibrium First-Order Melting and Second-Order Glass Transitions of the Vortex Matter in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$
• Similarity-Detection and Localization
• Exact Asymptotic Results for a Model of Sequence Alignment
• An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
• Explicit solution of the optimal fluctuation problem for an elastic string in random potential
• Cumulants of the current in the weakly asymmetric exclusion process
• Ground state energy of a non-integer number of particles with delta attractive interactions
• Interplay of Anisotropy and Disorder in the Doping-Dependent Melting and Glass Transitions of Vortices in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$
• Fluctuations of the winding number of a directed polymer in a random medium
• KPZ equation in one dimension and line ensembles
• Marginal pinning of vortices at high temperature

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