Dyson's Brownian Motion and Universal Dynamics of Quantum Systems

TL;DR

This paper links Dyson's Brownian motion to the universal dynamics of quantum systems, demonstrating a mathematical equivalence in eigenvalue distributions and correlations, with implications for understanding disordered quantum systems.

Contribution

It establishes a correspondence between eigenvalue evolution under Gaussian perturbations and Dyson's Fokker-Planck equation, confirming a conjecture about correlation functions in quantum systems.

Findings

Proves the equivalence between eigenvalue dynamics and Dyson's equation.

Shows the correspondence between quantum system correlations and Sutherland-Calogero-Moser system.

Identifies differences in multi-variable correlation functions between models.

Abstract

We establish a correspondence between the evolution of the distribution of eigenvalues of a $N\times N$ matrix subject to a random Gaussian perturbing matrix, and a Fokker-Planck equation postulated by Dyson. Within this model, we prove the equivalence conjectured by Altshuler et al between the space-time correlations of the Sutherland-Calogero-Moser system in the thermodynamic limit and a set of two-variable correlations for disordered quantum systems calculated by them. Multiple variable correlation functions are, however, shown to be inequivalent for the two cases.