[Paper Review] Elliptic law for real random matrices

Alexey Naumov|arXiv (Cornell University)|Jan 8, 2012

Random Matrices and Applications13 references17 citations

TL;DR

This paper establishes the Elliptic Law for real random matrices with i.i.d. off-diagonal entries $(X_{ij}, X_{ji})$ under finite fourth moment conditions. It proves that the empirical spectral distribution of $n^{-1/2} \mathbf{X}_n$ converges weakly in probability to a uniform distribution on an ellipse in the complex plane, whose axes are determined by the correlation $\rho = \mathbb{E}[X_{12}X_{21}]$, demonstrating universality independent of the underlying entry distribution.

ABSTRACT

In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i eq j}$ of entries. Under assumption of finite fourth moment of matrix entries it is proved that empirical spectral distribution of eigenvalues converges in probability to a uniform distribution on the ellipse. The axis of the ellipse are determined by correlation between $X_{12}$ and $X_{21}$. This result is called Elliptic Law. Limit distribution doesn't depend on distribution of matrix elements and the result in this sence is universal.

Motivation & Objective

• To establish the universality of the limiting spectral distribution for real non-Hermitian random matrices with dependent off-diagonal entries.
• To prove weak convergence in probability of the empirical spectral measure to a uniform distribution on an ellipse.
• To extend Girko's Elliptic Law beyond Gaussian assumptions to general i.i.d. matrix entries with finite fourth moments.
• To demonstrate that the limiting distribution depends only on the correlation $\rho = \mathbb{E}[X_{12}X_{21}]$, not on the specific distribution of entries.
• To provide a rigorous proof using logarithmic potential theory, small ball probability, and singular value analysis.

Proposed method

• Uses the logarithmic potential method to relate the spectral measure to the singular values of $n^{-1/2}\mathbf{X}_n - z\mathbf{I}$.
• Applies the $V$-transform to reduce the non-Hermitian problem to a Hermitian one involving singular values.
• Employs small ball probability estimates via the central limit theorem to control the least singular value.
• Decomposes the sphere to analyze invertibility and control the spectral norm of the resolvent.
• Establishes uniform integrability of the logarithm of the singular values to justify convergence of the spectral measure.
• Uses second-order expansion of the resolvent via Taylor’s formula to handle the dependence between $X_{ij}$ and $X_{ji}$.

Experimental results

Research questions

• RQ1 Does the empirical spectral distribution of real random matrices with i.i.d. off-diagonal pairs $(X_{ij}, X_{ji})$ converge to a universal limit under finite fourth moments?
• RQ2 How does the correlation $\rho = \mathbb{E}[X_{12}X_{21}]$ affect the shape of the limiting spectral ellipse?
• RQ3 Is the limiting distribution independent of the specific distribution of matrix entries, as long as fourth moments are finite?
• RQ4 Can the Elliptic Law be proven without assuming a density for the matrix entries, extending Girko’s original result?
• RQ5 What role does the least singular value play in the convergence of the spectral measure?

Key findings

• The empirical spectral distribution $\mu_n$ of $n^{-1/2}\mathbf{X}_n$ converges weakly in probability to a uniform distribution on an ellipse.
• The limiting measure $\mu$ has a constant density $g(x,y) = \frac{1}{\pi(1 - \rho^2)}$ inside the ellipse $\mathcal{E} = \left\{ x,y \in \mathbb{R} : \frac{x^2}{(1+\rho)^2} + \frac{y^2}{(1-\rho)^2} \leq 1 \right\}$.
• The shape of the ellipse is fully determined by the correlation $\rho = \mathbb{E}[X_{12}X_{21}]$, with semi-axes $1+\rho$ and $1-\rho$ along the real and imaginary axes.
• The limiting distribution is universal: it does not depend on the specific distribution of the matrix entries, only on $\rho$ and the finite fourth moment condition.
• The proof holds under minimal moment assumptions: $\mathbb{E}[X_{12}] = \mathbb{E}[X_{21}] = 0$, $\operatorname{Var}(X_{12}) = \operatorname{Var}(X_{21}) = 1$, and $\mathbb{E}[|X_{12}|^4], \mathbb{E}[|X_{21}|^4] \leq M_4$.
• The result is extended to complex asymmetric matrices, where the limiting ellipse has semi-axes $1+|\rho|$ and $1-|\rho|$ in orthogonal directions determined by the phase of $\rho$.