Tracy-Widom fluctuations in 2D random Schrodinger operators

TL;DR

This paper establishes Tracy-Widom fluctuations for the smallest eigenvalues of a 2D random Schrödinger operator on a hexagonal lattice, linking spectral properties to integrable polymer models and their free energy fluctuations.

Contribution

It introduces a novel connection between 2D random Schrödinger operators and integrable directed polymer models, proving Tracy-Widom fluctuations for eigenvalues.

Findings

Tracy-Widom fluctuations for the smallest eigenvalue.

Relation between eigenvalues and nonintersecting partition functions.

Connection to the log-Gamma polymer model.

Abstract

We construct a random Schrodinger operator on a subset of the hexagonal lattice and study its smallest positive eigenvalues. Using an asymptotic mapping, we relate them to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the random Schrodinger operator. We also relate the distribution of its k smallest positive eigenvalues to the nonintersecting partition functions of order k.