Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers

TL;DR

This paper derives the Tracy-Widom distribution for free energy fluctuations in 1D directed polymers using Bethe ansatz and quantum boson mappings, connecting statistical physics with random matrix theory.

Contribution

It provides a Bethe ansatz derivation of the Tracy-Widom distribution for directed polymers, linking integrable models to universal fluctuation statistics.

Findings

Distribution of free energy fluctuations follows Tracy-Widom distribution

Mapping to quantum bosons simplifies the problem

Reduction to Fredholm determinant with Airy kernel

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 a many body quantum boson system with attractive interactions. Performing the summation over the entire spectrum of excited states the problem is reduced to the Fredholm determinant with the Airy kernel which is known to yield the Tracy-Widom distribution