# Dyson’s disordered linear chain from a random matrix theory viewpoint

@article{Forrester2021DysonsDL,
title={Dyson’s disordered linear chain from a random matrix theory viewpoint},
author={Peter J. Forrester},
journal={Journal of Mathematical Physics},
year={2021}
}
• P. Forrester
• Published 7 January 2021
• Physics
• Journal of Mathematical Physics
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…
2 Citations
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## References

SHOWING 1-10 OF 67 REFERENCES
Impurity models and products of random matrices
• Mathematics, Computer Science
• 2016
The aim is to introduce the reader to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2×2 case, and state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.
The classical β-ensembles with β proportional to 1/N: From loop equations to Dyson’s disordered chain
• Mathematics
Journal of Mathematical Physics
• 2021
In the classical β-ensembles of random matrix theory, setting β = 2α/N and taking the N → ∞ limit gives a statistical state depending on α. Using the loop equations for the classical β-ensembles, we
Perturbation expansion for a one-dimensional Anderson model with off-diagonal disorder
The weak disorder expansion for a random Schrödinger equation with off-diagonal disorder in one dimension is studied. The invariant measure, the density of states, and the Lyapunov exponent are
Lyapunov exponents, one-dimensional Anderson localization and products of random matrices
• Mathematics
• 2013
The concept of Lyapunov exponent has long occupied a central place in the theory of Anderson localisation; its interest in this particular context is that it provides a reasonable measure of the
The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity
• Mathematics
• 2013
We study products of arbitrary random real 2×2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,ℝ), we identify a continuum regime where the mean values and the
Random discrete Schrödinger operators from random matrix theory
• Mathematics, Computer Science
• 2007
Random, discrete Schrodinger operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature β are investigated and it is shown that as a function of β they undergo a transition from a regime of (power-law) localized eigenstates with a pure point spectrum for β < 2 to a regime for extended states with a singular continuous spectrum.
Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model
• Physics
• 2015
We establish the connection between a multichannel disordered model --the 1D Dirac equation with $N\times N$ matricial random mass-- and a random matrix model corresponding to a deformation of the