Probability of all eigenvalues real for products of standard Gaussian matrices

Abstract

With $\{X_i\}$ independent $N\times N$ standard Gaussian random matrices, the probability $p_{N,N}^{P_m}$ that all eigenvalues are real for the matrix product $P_m=X_m{X}_{m-1}\cdots X_1$ is expressed in terms of an $N/2\times N/2$ ($N$ even) and $(N+1)/2\times (N+1)/2$ ($N$ odd) determinant. The entries of the determinant are certain Meijer $G$-functions. In the case $m=2$ high precision computation indicates that the entries are rational multiples of ${\pi }^2$, with the denominator a power of 2, and that to leading order in $N$ $p_{N,N}^{P_m}$ decays as $(\pi /4{)}^{N^2/2}$. We are able to show that for general $m$ and large $N$, $p_{N,N}^{P_m}\sim b_m^{N^2}$ with an explicit $b_m$. An analytic demonstration that $p_{N,N}^{P_m}\to 1$ as $m\to \infty$ is given.

15 pages; in version 2, $p_{N,N}^{P_m}\to 1$ as $m\to \infty$ is shown for general $N$, not just $N=2$ as in version 1

References in corpus (7)

• Elliptic integral evaluations of Bessel moments
• The Cauchy two-matrix model
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• The Lyapunov exponent of products of random $2\times 2$ matrices close to the identity
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)
• Products of random matrices and generalised quantum point scatterers
• Non-Hermitean Wishart random matrices (I)

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• Drift-diffusion on a Cayley tree with stochastic resetting: the localization-delocalization transition
• Eigenvector statistics of the product of Ginibre matrices
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Spectral statistics for the difference of two Wishart matrices
• Probability that product of real random matrices have all eigenvalues real tend to 1
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble
• Equilibrium measures for higher dimensional rotationally symmetric Riesz gases