Dynamics of an outlier in the Gaussian Unitary Ensemble

John Mateus; Gabriel Téllez; Frédéric van Wijland

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arXiv · 2509.14424

Dynamics of an outlier in the Gaussian Unitary Ensemble

John Mateus·Gabriel Téllez·Frédéric van Wijland

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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.

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John Mateus, Gabriel Téllez, Frédéric van Wijland. 2025-09-17. Dynamics of an outlier in the Gaussian Unitary Ensemble. https://arxiv.org/abs/2509.14424

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