Anthony Mays (2007-2011)


A geometrical triumvirate of real random matrices


The eigenvalue correlation functions for random matrix ensembles are fundamental descriptors of the statistical properties of these ensembles. In this work we present a five-step method for the calculation of these correlation functions, based on the method of (skew-) orthogonal polynomials. This scheme systemises existing methods and also involves some new techniques. By way of illustration we apply the scheme to the well known case of the Gaussian orthogonal ensemble, before moving on to the real Ginibre ensemble. A genealising parameter is then introduced to interpolate between the GOE and the real Ginibre ensemble. These real matrices have orthogonal symmetry, which is known to lead to Pfaffian or quaternion determinant processes, yet Pfaffian and quaternion determinants are not defined for odd-sized matrices. We present two methods for the calculation of the correlation functions in this case: the first is an extension of the even method, and the second establishes the odd case as a limit of the even case.

Having demonstrated our methods by reclaiming known results, we move on to study an ensemble of matrices \( Y = A^{-1} B \), where \( A \) and \( B \) are each real Ginibre matrices. This ensemble is known as the real spherical ensemble . By a convenient fractional linear transformation, we map the eigenvalues into the unit disk to obtain a rotationally invariant distribution of eigenvalues. The correlation functions are then calculated in terms of these new variables by means of finding the relevant skew orthogonal polynomials. The expected number real eigenvalues is computed, as is the probability of obtaining any number of real eigenvalues; the latter is compared to numerical simulation. The generating function is given by an explicit factorised polynomial, in which the zeros are gamma functions.

We show that in the limit of large matrix dimension, the eigenvalues (after stereographic projection) are uniformly distributed on the sphere, a result which is part of a universality result called the spherical law . By taking a different limit, we also show that the local behaviour of the eigenvalues matches that of the real Ginibre ensemble, which corresponds to the planar limit of the sphere.

Lastly, we examine the third ensemble in the triumvirate, the real truncated ensemble , which is formed by truncating \( L \) rows and columns from an \( N \times N \) Haar distributed orthogonal matrix. By applying the five-step scheme and by averaging over characteristic polynomials we proceed to calculate correlation functions and probabilities analogously to the other ensembles considered in this work. The probabilities of obtaining real eigenvalues are again compared to numerical simulation. In the large \(N\) limit (with small \(L\)) we find that the eigenvalues are uniformly distributed over the anti-sphere (after being suitably projected). This leads to a conjecture that, analogous to the circular law and spherical law, there exists an anti-spherical law. As we found for the spherical ensemble, we also find that in some limits the behaiour of this ensemble matches that of the real Ginibre ensemble.

Published papers:

A method to calculate correlation functions for \(\beta = 1\) random matrices of odd size (with P.J. Forrester), Journal of Statistical Physics 134, 443-462 (2011)

Pfaffian point process for the Gaussian real generalized eigenvalue problem (with P.J. Forrester), Probability Theory and Related Fields 154, 1-47 (2012)