Dynamics of an outlier in the Gaussian unitary ensemble
Phys. Rev. E 112, 064104 – Published 1 December, 2025
DOI: https://doi.org/10.1103/22b9-4q39
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-Péché transition. Amusingly, while the statics is well known and accessible via the Hikami-Brézin integrals, our approach for the dynamics is explicitly based on the use of orthogonal polynomials.
