Free Convolution and Generalized Dyson Brownian Motion
Abstract
The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of R-transforms for the sum in terms of a dynamical evolution of "particles" (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the "velocity" of the generalized Dyson motion. We extend our result to products of free matrices.
Cite
@article{arxiv.2412.03696,
title = {Free Convolution and Generalized Dyson Brownian Motion},
author = {Pierre Bousseyroux and Jean-Philippe Bouchaud},
journal= {arXiv preprint arXiv:2412.03696},
year = {2025}
}