Random Matrices in Non-confining Potentials

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 $a\ge a^*$, 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. We prove that this flux $j_a$ displays a second order phase transition at the critical value $a^*$ such that $j_a\sim C (a^*-a)^{3/2}$ when $a\uparrow a^*$ where $C$ is an explicit constant. In the subcritical 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 $x\rightarrow \pm \infty$. Our method applies to other non-confining potentials and we further investigate a family of quartic potentials, which were already studied in (Brezin et al. in Commun Math Phys 59:35–51, 1978) to count planar diagrams.

Appendices

Appendix A: Boltzmann Weight of the Hermitian Diffusion Process $H$

Let us check that the probability distribution $P$ defined in (2.1) is a stationary measure of the stochastic differential system (2.4). First notice that, if $M$ is a $N\times N$ real matrix, then the gradient of the function $M\rightarrow \mathrm{Tr}(V(M))\in \mathbb {R}$ with respect to the $N^2$ entries of the matrix $M$, is the function $M\rightarrow V'(M^\dagger )$. Thus, if $H$ is a Hermitian matrix, we simply have

$$\begin{aligned} \nabla \, \mathrm{Tr}(V(H)) = V'(H). \end{aligned}$$

(9.17)

It remains to check that the probability distribution $P$ is the unique stationary solution of the Fokker Planck equation satisfied by the (stationary) transition probability of the diffusion process $(H(t))$,

$$\begin{aligned} \frac{\partial P}{\partial t} = 0 = \frac{1}{2}\, \sum _{i,j=1} \frac{\partial }{\partial H_{ij}} \left[ \left( \nabla {\text {Tr}}(V(H))\right) _{ij} P(H) \right] + \frac{1}{2N} \sum _{i,j=1} \frac{1+ \delta _{i=j}}{2} \frac{\partial ^2}{\partial H_{ij}^2} P(H). \end{aligned}$$

(9.18)

The reader may actually check that the function $P$ as defined in (2.1) satisfies, for any Hermitian matrix $H$, the following conditions

$$\begin{aligned} \frac{\partial }{\partial H_{ii}} P(H)&=- N \, \left( \nabla {\text {Tr}}(V(H))\right) _{ii} P(H),\nonumber \\ \frac{\partial }{\partial H_{ij}} P(H)= \frac{\partial }{\partial H_{ji}} P(H)&= -2 N \, \left( \nabla {\text {Tr}}(V(H))\right) _{ij} P(H) \, \quad {\text{ if }} \quad i < j, \end{aligned}$$

(9.19)

under which (9.18) trivially holds. The factor $2$ which appears in the second line (9.19) is due to the symmetry of the matrix $H$.

Appendix B: Stieltjes Transform Properties

The Stieltjes transform is frequently used in RMT for the study of empirical spectral densities in the large $N$ limit.

A measure $\mu$ is characterized by its Stieltjes transform, which is an analytic function $G: \mathbb H \rightarrow \mathbb H$ ($\mathbb H$ denotes the open upper half-plane), defined as

$$\begin{aligned} G(z):=\int _\mathbb {R}\frac{\mu (dx)}{x-z}. \end{aligned}$$

We have the following inversion formula valid for any measure $\mu$ on $\mathbb {R}$,

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\int _x^y \mathfrak {I}\, G(\lambda +i \, \varepsilon ) \, d\lambda = \pi \, \mu (x,y) + \frac{\pi }{2}(\mu (\{y\})-\mu (\{x\})) , \end{aligned}$$

(9.20)

where $\mathfrak {I}z$ denotes the imaginary part of $z\in \mathbb C$.

When the Stieltjes transform $G(z)$ has a continuous extension to $\mathbb {R}\cup \mathbb H$, it is easy to check that $\mu$ admits a smooth density with respect to the Lebesgue measure.

If $\mu$ is a probability measure, its Stieltjes transform $G(iy)$ behaves as $-1/(iy)$ when $y$ goes to $+\infty$. Reciprocally, Akhiezer’s theorem [35], page 93] states a useful criterium characterizing Stieltjes transforms of probability measure: $G$ is the Stieltjes transform of a probability measure if and only if $G$ is analytic on $\mathbb H$ with $G( \mathbb H ) \subseteq \mathbb H$ and $G(iy) \sim -1/(iy)$ as $y \rightarrow + \infty$.