Truncations of Random Orthogonal Matrices

Abstract

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.

4 pages, final revised version (one reference added, minor changes in Introduction)

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Cited by in corpus (26)

• Recent exact and asymptotic results for products of independent random matrices
• Weak Commutation Relations and Eigenvalue Statistics for Products of Rectangular Random Matrices
• Induced Ginibre ensemble of random matrices and quantum operations
• Universal microscopic correlation functions for products of truncated unitary matrices
• Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials
• X-shaped and Y-shaped Andreev resonance profiles in a superconducting quantum dot
• Analogies between random matrix ensembles and the one-component plasma in two-dimensions
• New spectral relations between products and powers of isotropic random matrices
• The limiting Kac random polynomial and truncated random orthogonal matrices
• Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
• Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices
• A real quaternion spherical ensemble of random matrices
• The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices
• Matrix models and eigenvalue statistics for truncations of classical ensembles of random unitary matrices
• On the number of real eigenvalues of a product of truncated orthogonal random matrices
• A geometrical triumvirate of real random matrices
• Clustering of periodic orbits and ensembles of truncated unitary matrices
• Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations
• Real spectra of large real asymmetric random matrices
• Expansion of polynomial Lie group integrals in terms of certain maps on surfaces, and factorizations of permutations
• Random sparse generators of Markovian evolution and their spectral properties
• Real eigenvalues of elliptic random matrices
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Phase transitions for infinite products of large non-Hermitian random matrices
• Truncations of random symplectic unitary matrices
• How many eigenvalues of a product of truncated orthogonal matrices are real?