The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Abstract

We study the product $P_m$ of $m$ real Ginibre matrices with Gaussian elements of size $N$, which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when $m\to \infty$ at fixed $N$. In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when $N\to \infty$ and $m$ is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at $m=1$ prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, $m=\alpha N$. We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when $\alpha \to \infty$ and $\alpha \to 0$, respectively.

arXiv comment22 pages, 4 figures

References in corpus (19)

• Eigenvalue statistics of the real Ginibre ensemble
• Directed Quantum Chaos
• Recent exact and asymptotic results for products of independent random matrices
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Determinantal processes with number variance saturation
• General Eigenvalue Correlations for the Real Ginibre Ensemble
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Lyapunov exponents for products of complex Gaussian random matrices
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• Probability of all eigenvalues real for products of standard Gaussian matrices
• On the largest Lyapunov exponent for products of Gaussian matrices
• On the number of real eigenvalues of products of random matrices and an application to quantum entanglement
• Universality of local spectral statistics of products of random matrices
• Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices
• Real eigenvalues of non-Gaussian random matrices and their products
• On the number of real eigenvalues of a product of truncated orthogonal random matrices
• Probability that product of real random matrices have all eigenvalues real tend to 1

Cited by in corpus (1)

• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble