Mutually avoiding paths in random media and largests eigenvalues of random matrices

Abstract

Recently, it was shown that the probability distribution function (PDF) of the free energy of a single continuum directed polymer (DP) in a random potential, equivalently of the height of a growing interface described by the Kardar-Parisi-Zhang (KPZ) equation, converges at large scale to the Tracy-Widom distribution.The latter describes the fluctuations of the largest eigenvalue of a random matrice, drawn from the Gaussian Unitary Ensemble (GUE), and the result holds for a DP with fixed endpoints, i.e. for the KPZ equation with droplet initial conditions. A more general conjecture can be put forward, relating the free energies of $N>1$ non-crossing continuum DP in a random potential, to the $N$-th largest eigenvalues of the GUE. Here, using replica methods, we provide an important test of this conjecture by calculating exactly the right tails of both PDF's and showing that they coincide for arbitrary $N$.

5 pages, 2 figures + 4 pages of supplementary material

References in corpus (11)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Burgers Turbulence
• Exact scaling functions for one-dimensional stationary KPZ growth
• An exact solution for the KPZ equation with flat initial conditions
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Correlation functions of the one-dimensional attractive Bose gas
• Exact distribution of the maximal height of p vicious walkers
• Cumulants of the current in the weakly asymmetric exclusion process
• Unbinding transition in semi-infinite two-dimensional localized systems
• Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz