Consecutive Minors for Dyson's Brownian Motions

TL;DR

This paper demonstrates that the spectra of two consecutive principal minors in Dyson's Brownian motion form a Markov process, revealing new interlacing diffusion structures for beta=1, 2, 4, but not for three minors at certain beta values.

Contribution

It establishes that interlacing spectra of consecutive minors form a Markov diffusion process with explicit generator and invariant measure for beta=1, 2, 4, extending Dyson's original dynamics.

Findings

Spectra of two consecutive minors form a Markov process.

Explicit formulas for generator, transition probability, and invariant measure.

Spectra of three minors are not Markovian for beta=2, 4.

Abstract

In 1962, Dyson introduced dynamics in random matrix models, in particular into GUE (also for beta=1 and 4), by letting the entries evolve according to independent Ornstein-Uhlenbeck processes. Dyson shows the spectral points of the matrix evolve according to non-intersecting Brownian motions. The present paper shows that the interlacing spectra of two consecutive principal minors form a Markov process (diffusion) as well. This diffusion consists of two sets of Dyson non-intersecting Brownian motions, with a specific interaction respecting the interlacing. This is revealed in the form of the generator, the transition probability and the invariant measure, which are provided here; this is done in all cases: beta=1,~2,~4. It is also shown that the spectra of three consecutive minors ceases to be Markovian for \beta=2,~4.