Real eigenvalue statistics for products of asymmetric real Gaussian matrices
arXiv:1608.04097
Abstract
Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these - the eigenvalue density of the real eigenvalues - gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.
arXiv comment25 pages, 1 figure, to appear in Linear Algebra and its Applications
References in corpus (19)
- Eigenvalue statistics of the real Ginibre ensemble
- The Ginibre ensemble of real random matrices and its scaling limits
- Spectrum of the Product of Independent Random Gaussian Matrices
- Recent exact and asymptotic results for products of independent random matrices
- Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
- The Complex Laguerre Symplectic Ensemble of Non-Hermitian Matrices
- Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
- Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble
- Eigenvalue statistics for product complex Wishart matrices
- On the Asymptotic Spectrum of Products of Independent Random Matrices
- Pfaffian formulae for one dimensional coalescing and annihilating systems
- Wigner-Poisson statistics of topological transitions in a Josephson junction
- Products of Independent Gaussian Random Matrices
- Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for product of two Gaussian matrices
- Real eigenvalues of non-Gaussian random matrices and their products
- A geometrical triumvirate of real random matrices
- Rank probabilities for real random tensors
- Probability that product of real random matrices have all eigenvalues real tend to 1
- Statistics of Extreme Spacings in Determinantal Random Point Processes