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

TL;DR

This paper investigates the connection between the free energy distributions of multiple non-crossing continuum directed polymers in random media and the largest eigenvalues of random matrices, providing exact tail calculations to test a conjecture.

Contribution

It offers a rigorous test of a conjecture linking multiple directed polymers' free energies to the top eigenvalues of GUE matrices using replica methods.

Findings

Exact calculation of the right tails of free energy PDFs for multiple polymers.

Confirmation that the distributions coincide with the largest eigenvalues of GUE matrices.

Supports the conjecture relating non-crossing polymers to GUE eigenvalues.

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 $N$.