Weakly universal dynamical correlations between eigenvalues of large random matrices

TL;DR

This work analyzes how universal features of eigenvalue correlations in large random matrices persist under Dyson Brownian motion. By formulating a fluctuating hydrodynamics for the resolvent and applying Macroscopic Fluctuation Theory, it derives a central dynamical relation for the resolvent correlations: $\Gamma(z,z';t) = -\frac{1}{\beta N^2}\frac{Q(\zeta)}{Q(z)}\frac{a^2- \zeta z'+\sqrt{\zeta^2-a^2}\sqrt{z'^2-a^2}}{\sqrt{z^2-a^2}\sqrt{z'^2-a^2}(\zeta-z')^2}$ with $\frac{d\zeta}{dt}=Q(\zeta)\sqrt{\zeta^2-a^2}$ and $\zeta(0)=z$, showing that dynamical correlations retain a large degree of universality through a characteristic trajectory. The results are worked out in detail for harmonic and quartic potentials, revealing explicit time-dependent forms and long-time decays, and are extended to higher-degree potentials where multiple relaxation channels emerge. The framework paves the way for dynamical large-deviation analyses and extensions to other ensembles, connecting stochastic eigenvalue dynamics to fluctuating hydrodynamics and dynamic free-probability constructs via the resolvent.

Abstract

It was shown roughly thirty years ago that the density correlations of eigenvalues of large random matrices display a universal form, independent of most of the details of the distribution of the random matrix itself. We show that when the matrix elements evolve according to a Dyson Brownian motion, dynamical correlations retain a large degree of the universality found at equal times when expressed in terms of the characteristics of some partial differential equation in the complex plane.

Figures (2)

• Figure 1: Trajectories of the characteristics starting from right below the branch cut $[-a,a]$ for a quartic potential $V(\lambda)=\frac{\lambda^4}{4}$ ($Q(\lambda)=\lambda^2+a^2/2$ and $a^2=2\sqrt{2/3}$). All flow to the $z_1=-ia/\sqrt{2}$ root of $Q$ lying in the lower half plane (the initial values are $-0.7$ (green), $0.5$ (blue) and $1$ (orange)).
• Figure 2: Trajectories of the characteristics starting from right below the branch cut $[-a,a]$ for a potential $V(\lambda)=\frac{\lambda^8}{8}$ ($Q(\lambda)=\lambda^6+\frac{a^2}{2}\lambda^4+\frac{3}{8}a^4 \lambda^2+\frac{5}{16}a^6$ and $a^8=64/35$). The roots are $z_1\simeq-0.9 i$, $z_2=0.7-0.6 i$ (and $z_3=-z_2^*$) and $a\simeq 1.1$. The characteristics are initialized with positive reals parts (due to the symmetries of the potential) at the following (just below the real axis): $z=0.1$ (blue), $z=0.5$ (orange), $z=0.7$ (green), $z=0.9$ (red) and $z=1.05$ (violet). The basins of attraction of each $z_1$ and $z_2$ are non trivial.