Random matrices in non-confining potentials

TL;DR

This paper studies invariant matrix diffusions in non-confining cubic potentials, revealing a phase transition in spectral density and constructing solutions that restart after explosions, with implications for other non-confining potentials.

Contribution

It introduces a method to construct and analyze invariant matrix processes in non-confining potentials, characterizing spectral dynamics and phase transitions, including explicit stationary states and tail behaviors.

Findings

Spectral density exhibits a phase transition at a critical parameter value.

For parameters above the critical value, eigenvalues are confined with compact support.

Below the critical value, eigenvalues have a stationary flux with heavy tails.

Abstract

We consider invariant matrix processes diffusing in non-confining cubic potentials of the form $V_a(x)= x^3/3 - a x, a\in \mathbb{R}$. We construct the trajectories of such processes for all time by restarting them whenever an explosion occurs, from a new (well chosen) initial condition, insuring continuity of the eigenvectors and of the non exploding eigenvalues. We characterize the dynamics of the spectrum in the limit of large dimension and analyze the stationary state of this evolution explicitly. We exhibit a sharp phase transition for the limiting spectral density $\rho_a$ at a critical value $a=a^*$. If $a\geq a^*$, then the potential $V_a$ presents a well near $x=\sqrt{a}$ deep enough to confine all the particles inside, and the spectral density $\rho_a$ is supported on a compact interval. If $a<a^*$ however, the steady state is in fact dynamical with a macroscopic stationary flux of particles flowing across the system. In this regime, the eigenvalues allocate according to a stationary density profile $\rho_{a}$ with full support in $\mathbb{R}$, flanked with heavy tails such that $\rho_{a}(x)\sim C_a /x^2$ as $x\to \pm \infty$. Our method applies to other non-confining potentials and we further investigate a family of quartic potentials, which were already studied in Br\'ezin et al. to count planar diagrams.