Dynamics of an outlier in the Gaussian Unitary Ensemble
John Mateus, Gabriel Téllez, Frédéric van Wijland
Published: 2025/9/17
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.