Dynamics of an outlier in the Gaussian Unitary Ensemble

TL;DR

The work addresses the dynamical Baik–Ben Arous–Péché transition in a Gaussian Unitary Ensemble by imposing Dyson Brownian motion on a matrix with a single outlier. It solves the $\beta=2$ case exactly through a Darboux-transformed, free-fermion framework that yields a determinantal representation with a Hermite-based kernel, and then analyzes large-$N$ scaling to separate bulk and outlier contributions, revealing a drifting outlier and an Airy-edge regime at the transition. The paper provides explicit formulas for the outlier center $\lambda^*(t)=\lambda_0+t/\lambda_0$, the early-time variance $\sigma_\lambda^2= t(\lambda_0^2-t)/(N\lambda_0^2)$, and the Airy scaling at $t^*=\lambda_0^2$, together with a detailed numerical validation. For $\beta\neq 2$, it presents a mean-field, Brownian-bridge-type conjecture for the outlier dynamics and variance, supported by simulations, suggesting universality of the dynamical BBP transition beyond the solvable case.

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.

Figures (7)

• Figure 1: Illustration of the different regions in the Rotach--Plancherel asymptotics of the Hermite polynomials (for $N=50$). The black curve represents a typical term contributing to the bulk, whereas the blue curve shows the outlier contribution. Both are plotted as functions of $x = \lambda / \sqrt{4t}$. Each term has a different amplitude and has been rescaled by its respective maximum to allow comparison on the same plot. The outlier (in blue) is centered around $x^* = x_0 + (4x_0)^{-1}$, where $x_0 = \lambda_0 / \sqrt{4t}$ corresponds to the initial eigenvalue $\lambda_0$. As time $t$ goes by, $x^*$ moves from the exponential decay zone to the oscilatory zone, as indicated by the red arrow.
• Figure 2: Snapshots of the eigenvalue density profile for increasing times from left to right.
• Figure 3: Bulk contribution (black), outlier contribution (blue dashed), and total density (red) at the critical time $t^*$ in the Airy scaling zone. The variable $s$ denotes the rescaled position defined in Eq. \ref{['eq:def-s']}.
• Figure 4: Time evolution from a single simulation (left) and the eigenvalue distribution aggregated over all simulations (right), using parameters $\beta=2$, $N=100$ eigenvalues, and $\lambda_0=0.1$ (shown as the black line in the left plot). Note that the eigenvalue trajectories do not intersect due to the adaptive time step defined in Eq. \ref{['eq:timestep']}. Green line indicates the time $t^*$ given by Eq. \ref{['eq:outlier-catch']}, when the bulk reaches the outlier. On the right, the probability distribution of the eigenvalues averaged over all simulations is shown at different times, that are marked with vertical lines in the left figure using the same color for each. Red and yellow lines highlight the separation between the bulk and the outlier for times $t<t^*$. As time progresses, the bulk catches up to the outlier (green line). After this point, the distinction between the bulk and the outlier becomes unclear (blue line).
• Figure 5: Evolution of the averaged outlier position (blue dots) for $N=100$ eigenvalues, $\beta=2$, and $\lambda_0=0.1$, compared with the theoretical prediction (red line) given by Eq. (\ref{['eq:outlier-eq']}). Also shown is the evolution of the averaged edge of the bulk (purple dots), determined by tracking the eigenvalue closest to the outlier, averaged over all simulations. The bulk density predicted by the Wigner semicircle law is shown as a solid black line. Shaded areas represent the standard deviations of the simulated outlier and bulk positions, respectively. The inset displays the time evolution over the full simulation time domain.