Dyson's disordered linear chain from a random matrix theory viewpoint

TL;DR

This paper reviews Dyson's original work on disordered linear chains, highlighting an exact solution for a specific disorder distribution, its universal features, and connections to random matrix theory and Gaussian beta-ensembles.

Contribution

It provides a detailed analysis of Dyson's disordered chain using random matrix theory, revealing universal spectral features and linking to Gaussian beta-ensembles.

Findings

Exact solution shows a universal singularity in the density of states.

Universal features of weak disorder expansion near the band edge.

Connection established between the chain model and Gaussian beta-ensembles.

Abstract

The first work of Dyson relating to random matrix theory, "The dynamics of a disordered linear chain", is reviewed. Contained in this work is an exact solution of a so-called Type I chain in the case of the disorder variables being given by a gamma distribution. The exact solution exhibits a singularity in the density of states about the origin, which has since been shown to be universal for one-dimensional tight binding models with off diagonal disorder. We discuss this context and also point out some universal features of the weak disorder expansion of the exact solution near the band edge. Further, a link between the exact solution, and a tridiagonal formalism of anti symmetric Gaussian $\beta$-ensembles with $\beta$ proportional to $1/N$, is made.