Random Matrices: Theory and Applications

The purpose of this paper is to put forward the claim that Hurwitz’s paper [Über die Erzeugung der invarianten durch integration, Nachr. Ges. Wiss. Göttingen 1897 (1897) 71–90.] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups $SO(N)$ and $U(N)$. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrization — Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over $SO(N)$ and $U(N)$. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz’s ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a $2\times 2$ complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij}\sim {𝒩}(0,1)$. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability 26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{ℝ}}$ are the real eigenvalues of $G$, then for any even polynomial function $P(x)$ and even $N=2n$, we have the convergence in distribution to a normal random variable

We consider the generalized Jacobi weight $x^{\alpha }{(1-x)}^{\beta }|x-t{|}^{\gamma },x\in [0,1],t(t-1)>0$, $\alpha >-1,\beta >-1,\gamma \in ℝ$. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the $\tau$-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near $0,\infty$ and $1$ for generic $(\alpha ,\beta ,\gamma )$. For four special cases of $(\alpha ,\beta ,\gamma )$ which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694], and thus give another characterization of the Hankel determinant.

This work is a companion paper of [F. Gamboa, J. Nagel and A. Rouault, Sum rules via large deviations, J. Funct. Anal. 270 (2016) 509–559] and [F. Gamboa, J. Nagel and A. Rouault, Sum rules and large deviations for spectral matrix measures, preprint (2016), arXiv:1601.08135] (see also [J. Breuer, B. Simon and O. Zeitouni, Large deviations and sum rules for spectral theory — A pedagogical approach, to appear in J. Spectr. Theory, preprint (2016), arXiv:1608.01467]). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two-matrix models. The first one is the Gross–Witten (GW) ensemble. In the gapped regime, we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua–Pickrell (HP) ensemble. Unlike the GW ensemble the potential is here infinite at one point. Surprisingly, but as in [F. Gamboa, J. Nagel and A. Rouault, Sum rules via large deviations, J. Funct. Anal. 270 (2016) 509–559], we obtain a completely new sum rule for the deviation to the equilibrium measure of the HP ensemble. The case of spectral matrix measures is also studied. Indeed, in the case of HP ensemble, we extend our earlier works on large deviation for spectral matrix measure [F. Gamboa, J. Nagel and A. Rouault, Sum rules and large deviations for spectral matrix measures, preprint (2016), arXiv:1601.08135] and get here also a completely new sum rule.