Dynamics of an outlier in the Gaussian Unitary Ensemble
Statistical Mechanics
2025-11-18 v2
Abstract
We endow the elements of a random matrix drawn from the Gaussian Unitary Ensemble with a Dyson Brownian motion dynamics. We initialize the dynamics of the eigenvalues with all of them lumped at the origin, but one outlier. We solve the dynamics exactly which gives us a window on the dynamical scaling behavior at and around the Baik-Ben Arous-P\'ech\'e transition. Amusingly, while the statics is well-known and accessible via the Hikami-Br\'ezin integrals, our approach for the dynamics is explicitly based on the use of orthogonal polynomials.
Keywords
Cite
@article{arxiv.2509.14424,
title = {Dynamics of an outlier in the Gaussian Unitary Ensemble},
author = {John Mateus and Gabriel Téllez and Frédéric van Wijland},
journal= {arXiv preprint arXiv:2509.14424},
year = {2025}
}