Local central limit theorem for real eigenvalue fluctuations of elliptic GinOE matrices

TL;DR

This paper proves a local central limit theorem for the fluctuations in the number of real eigenvalues of elliptic GinOE matrices, extending previous results by analyzing the generating function's zeros and variance growth.

Contribution

It establishes a local CLT for real eigenvalue counts in elliptic GinOE matrices, using properties of the generating function and variance analysis, which was not previously known.

Findings

The generating function has only negative real zeros.

Variance of eigenvalue counts tends to infinity as matrix size grows.

The local CLT provides detailed fluctuation behavior of real eigenvalues.

Abstract

Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and real anti-symmetric, real Gaussian random matrices and controlled by a parameter $\tau$. Our interest is in the fluctuations of the number of real eigenvalues, for fixed $\tau$ when the expected number is proportional to the square root of the matrix size $N$, and for $\tau$ scaled to the weakly non-symmetric limit, when the number of eigenvalues is proportional to $N$. By establishing that the generating function for the probabilities specifying the distribution of the number of real eigenvalues has only negative real zeros, and using too the fact that variances in both circumstancesof interest tends to infinity as $N \to \infty$, the known central limit theorem for the fluctuations is strengthened to a local central limit theorem, and the rate of convergence is discussed.