[Paper Review] Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

Mohamed Bouali|arXiv (Cornell University)|Aug 30, 2014

Random Matrices and Applications5 references3 citations

TL;DR

This paper derives the exact asymptotic density of positive eigenvalues for the generalized Gaussian unitary ensemble (GGUE) with a determinant-weighted measure, showing that the probability all eigenvalues are positive decays as $\sim \exp[-\beta\theta(\alpha)n^2]$, where $\theta(\alpha)$ is explicitly computed. The eigenvalue density exhibits an inverse square-root singularity at the edge of the support, generalizing the Wigner semicircle law to ensembles conditioned on eigenvalue positivity.

ABSTRACT

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[σ, +\infty[$. In particular, we show that the probability that all the eigenvalues of an $(n imes n)$ random matrix are positive (negative) decreases for large $n$ as $\sim exp[-βθ(α)n^2]$ where the Dyson index $β$ characterizes the ensemble, $α$ is some extra parameter and the exponent $θ(α)$ is a function of $α$ which will be given explicitly. For $α=0$, $θ(0)= (\log 3)/4 = 0.274653...$ is universal. We compute the probability that the eigenvalues lie in the interval $[σ,+\infty[$ with $(σ>0,\; { m if}\;α>0)$ and $(σ\in\mathbb R,\; { m if }\;α=0)$. This generalizing the celebrated Wigner semicircle law to these restricted ensembles. It is found that the density of eigenvalues generically exhibits an inverse square-root singularity at the location of the barriers. These results generalized the case of Gaussian random matrices ensemble studied in \cite{D}, \cite{S}.

Motivation & Objective

• To compute the exact asymptotic statistical density of eigenvalues in the generalized Gaussian unitary ensemble (GGUE) conditioned on all eigenvalues being positive.
• To determine the large-$n$ decay rate of the probability that all $n \times n$ eigenvalues are positive, generalizing the known $\sim \exp(-\theta(0)n^2)$ result for standard GUE.
• To derive the limiting eigenvalue density for the conditioned ensemble, showing it differs from the Wigner semicircle law and exhibits an inverse square-root singularity at the edge.
• To generalize the Wigner semicircle law to ensembles with a determinant-weighted measure, where the eigenvalue support is restricted to $[a,b]$ with $a > 0$ for $\alpha > 0$.

Proposed method

• The analysis uses orthogonal polynomials on the positive real line with a weight $x^{2\mu_n} e^{-x^2}$, where $\mu_n \approx \alpha n$, to model the eigenvalue density.
• The eigenvalue density is expressed as $f_n(x) = \frac{1}{\sqrt{n}} \sum_{k=0}^{n-1} \varphi_k^{\mu_n}(\sqrt{n}x)^2$, with $\varphi_k^{\mu_n}$ being normalized truncated Hermite polynomials.
• The asymptotic behavior of the density is derived via a Riemann-Hilbert approach and integral representations, leading to a limiting density $f_{\alpha,a}(x)$ for $\alpha \geq 0$.
• The support $[a,b]$ is determined by solving a system of nonlinear equations involving $\alpha$, $a$, and $b$, which ensures the correct normalization and edge behavior.
• The method incorporates logarithmic potential theory and the use of generating functions to compute the equilibrium measure under the constraint of eigenvalue positivity.
• The decay exponent $\theta(\alpha)$ is derived from the variational problem of minimizing the energy functional under the constraint $\mu_n \approx \alpha n$, leading to an explicit expression for $\theta(\alpha)$.

Experimental results

Research questions

• RQ1 What is the exact asymptotic decay rate of the probability that all eigenvalues of an $n \times n$ matrix in the generalized Gaussian unitary ensemble are positive, as $n \to \infty$?
• RQ2 How does the eigenvalue density in the GGUE change when conditioned on all eigenvalues being positive, and how does it differ from the Wigner semicircle law?
• RQ3 What is the functional form of the limiting eigenvalue density $f_{\alpha,a}(x)$ for $\alpha > 0$, and how do its parameters $a$ and $b$ depend on $\alpha$?
• RQ4 What is the behavior of the eigenvalue density near the edge of the support, and does it exhibit a universal inverse square-root singularity?
• RQ5 How does the decay exponent $\theta(\alpha)$ vary with the parameter $\alpha$, and what is its value for $\alpha = 0$?

Key findings

• The probability that all eigenvalues are positive decays as $\sim \exp[-\beta\theta(\alpha)n^2]$ for large $n$, with $\theta(\alpha)$ explicitly computed as a function of $\alpha$.
• For $\alpha = 0$, the decay exponent is $\theta(0) = \frac{1}{4}\log 3 \approx 0.274653$, which is universal across all Dyson indices $\beta$.
• For small $\alpha \leq 0.34$, the exponent behaves as $\theta(\alpha) = \frac{1}{4}\log 3 - C\alpha + o(\alpha)$, with $C \approx 0.3482$, showing a linear correction to the universal value.
• The limiting eigenvalue density for $\alpha > 0$ is $f_{\alpha,a}(x) = \frac{1}{2\pi}\sqrt{\frac{b-x}{x-a}}\left(2x + b - a - 2\alpha\sqrt{\frac{a}{b}} \frac{1}{x}\right)$, with $a > 0$, $b > a$ determined by a system of equations.
• The support $[a,b]$ of the density is strictly positive for $\alpha > 0$, and the density exhibits an inverse square-root singularity at both endpoints $a$ and $b$, indicating edge effects.
• For $\alpha = 0$, the density reduces to $f_{0,0}(x) = \frac{1}{2\pi}\sqrt{\frac{b-x}{x}}(2x + b)$ with $b = \frac{2}{3}\sqrt{6}$, and the support is $[0, \frac{2}{3}\sqrt{6}]$, generalizing the Wigner semicircle law to the positive eigenvalue condition.