Consecutive Minors for Dyson's Brownian Motions
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.
Keywords
Cite
@article{arxiv.1007.0220,
title = {Consecutive Minors for Dyson's Brownian Motions},
author = {Mark Adler and Eric Nordenstam and Pierre van Moerbeke},
journal= {arXiv preprint arXiv:1007.0220},
year = {2012}
}
Comments
32 pages, 1 figure