A method to calculate correlation functions for $\beta =1$ random matrices of odd size

Abstract

The calculation of correlation functions for $\beta =1$ random matrix ensembles, which can be carried out using Pfaffians, has the peculiar feature of requiring a separate calculation depending on the parity of the matrix size N. This same complication is present in the calculation of the correlations for the Ginibre Orthogonal Ensemble of real Gaussian matrices. In fact the methods used to compute the $\beta =1$, N odd, correlations break down in the case of N odd real Ginibre matrices, necessitating a new approach to both problems. The new approach taken in this work is to deduce the $\beta =1$, N odd correlations as limiting cases of their N even counterparts, when one of the particles is removed towards infinity. This method is shown to yield the correlations for N odd real Gaussian matrices.

20 pages, corrected typos

References in corpus (10)

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• Eigenvalue statistics of the real Ginibre ensemble
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• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Moments of random matrices and hypergeometric orthogonal polynomials
• Matrix averages relating to the Ginibre ensembles
• Exact and asymptotic features of the edge density profile for the one component plasma in two dimensions
• The limiting Kac random polynomial and truncated random orthogonal matrices
• Correlation Functions for β=1 Ensembles of Matrices of Odd Size
• Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
• Finite size corrections for real eigenvalues of the elliptic Ginibre matrices
• The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$
• Real eigenvalues of elliptic random matrices
• The probability of almost all eigenvalues being real for the elliptic real Ginibre ensemble
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Preserving Topology while Breaking Chirality: From Chiral Orthogonal to Anti-symmetric Hermitian Ensemble