Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers

TL;DR

This paper derives the Tracy-Widom distribution for free energy fluctuations in 1D directed polymers by mapping the problem to a quantum boson system and summing over all states, revealing universal statistical behavior.

Contribution

It provides a replica Bethe ansatz derivation of the Tracy-Widom distribution for directed polymers, connecting quantum many-body solutions to universal fluctuation statistics.

Findings

Derivation of the full eigenfunctions and eigenvalues of the quantum boson system.

Reduction of the problem to a Fredholm determinant with the Airy kernel.

Confirmation that free energy fluctuations follow the Tracy-Widom distribution.

Abstract

The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions. We find the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.