The limiting Kac random polynomial and truncated random orthogonal matrices

Abstract

An exact calculation of the eigenvalue statistics of truncated random Haar distributed real orthogonal matrices has recently been carried out by Khoruzhenko, Sommers and Zyczkowski. We further develop this calculation, and use it to deduce a Pfaffian form of the correlations for the zeros of the limiting Kac random polynomial. This contrasts with the forms known from previous studies of the real zeros (a multidimensional Gaussian integral with the integrand multiplied by the absolute values of the variables) and the complex zeros (a Hafnian).

12 pages, no figures

References in corpus (7)

• Truncations of Random Orthogonal Matrices
• General Eigenvalue Correlations for the Real Ginibre Ensemble
• Zeros of Random Analytic Functions
• Characteristic polynomials in real Ginibre ensembles
• A method to calculate correlation functions for $\beta =1$ random matrices of odd size
• The chiral Gaussian two-matrix ensemble of real asymmetric matrices
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices

Cited by in corpus (9)

• 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
• Correlation functions for zeros of a Gaussian power series and Pfaffians
• 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
• Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems
• Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations
• Geometry of the Ising persistence problem and the universal Bonnet-Manin Painlevé VI distribution