Random Matrices: Theory and Applications

Roy’s largest root is a common test statistic in multivariate analysis, statistical signal processing and related fields. According to Anderson [An Introduction to Multivariate Statistical Analysis, 3rd edn. (Wiley, New York, 2003)], it is numerically known that compared with the other three tests of linear hypotheses, Roy’s largest root test has the highest power under rank-one alternatives. Therefore, it is important to study the asymptotic distribution of the largest root under rank-one alternatives to obtain an estimation of the power. To the best of our knowledge, no one had solved the problem until Johnstone and Nadler [Roy’s largest root test under rank-one alternatives, Biometrika 104(1) (2017) 181–193] presented a tractable approximation of the distribution of Roy’s largest root test where the alternative is of rank one and the variance of the noise tends to zero. It is natural to ask how Roy’s largest root test performs under other alternatives, for example, rank-finite alternatives. Therefore, we are more interested in the power estimates of Roy’s largest root test under wider alternatives, namely, whether its power is still higher than that of the other three tests of linear hypotheses. In fact, the distribution of the largest root under rank-finite alternatives can be characterized as the distribution of the largest sample eigenvalue among several spiked models. In this paper, we employ the asymptotic results of the spiked eigenvalues derived by Bai and Yao [Central limit theorems for eigenvalues in a spiked population model, Ann. Inst. Henri. Poincarè Probab. Statist. 44(3) (2008) 447–474; On sample eigenvalues in a generalized spiked population model, J. Multivar. Anal. 106 (2012) 167–177] and Wang and Yao [Extreme eigenvalues of large-dimensional spiked Fisher matrices with application, Ann. Statist. 45 (2017) 415–460] to obtain the approximate power of Roy’s largest root test under a high-dimensional setting; more importantly, our results are distribution-free and applicable to cases involving rank-finite alternatives.

The aim of this paper is to give fine asymptotics for random variables with moments of Gamma type. Among the examples, we consider random determinants of Laguerre and Jacobi beta ensembles with varying dimensions (the number of observed variables and the number of measurements vary and may be different). In addition to the Dyson threefold way of classical random matrix models (GOE, GUE, GSE), we study random determinants of random matrices of the so-called tenfold way, including the Bogoliubov–de Gennes and chiral ensembles from mesoscopic physics. We show that fixed-trace matrix ensembles can be analyzed as well. Finally, we add fine asymptotics for the $p(n)$-dimensional volume of the simplex with $p(n)+1$ points in ${ℝ}^n$ distributed according to special distributions, which is strongly correlated to Gram matrix ensembles. We use the framework of mod-$\phi$ convergence to obtain extended limit theorems, Berry–Esseen bounds, precise moderate deviations, large and moderate deviation principles as well as local limit theorems. The work is especially based on the recent work of Dal Borgo et al. [Mod-Gaussian convergence for random determinants, Ann. Henri Poincaré (2018)].

We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [Sum rules via large deviations, J. Funct. Anal. 270(2) (2016) 509–559] for spectral measures of classical ensembles (Gauss–Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [Sum rules and large deviations for spectral matrix measures, Bernoulli 25(1) (2018) 712–741]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten–McKay law. As an important intermediate step, we derive the distribution of matricial canonical moments of the Jacobi ensemble.

This paper considers inhomogeneous Erdős–Rényi random graphs ${𝔾}_N$ on $N$ vertices in the non-sparse non-dense regime. The edge between the pair of vertices $\{i,j\}$ is retained with probability ${𝜀}_{N}f(\frac{i}{N},\frac{j}{N})$, $1\le i\ne j\le N$, independently of other edges, where $f:[0,1]\times [0,1]\to [0,\infty )$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y\in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix ${\mathrm{\Delta }}_N$ associated with ${𝔾}_N$, in the limit as $N\to \infty$ when ${lim}_{N\to \infty }{𝜀}_N=0$ and ${lim}_{N\to \infty }N{𝜀}_N=\infty$. In particular, we show that the empirical spectral distributions of $A_N$ and ${\mathrm{\Delta }}_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y)=r(x)r(y)$ with $r:[0,1]\to [0,\infty )$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, we apply our results to constrained random graphs, Chung–Lu random graphs and social networks.

We present new expressions for the densities and distributions of the largest eigenvalue and the trace of a Beta–Wishart matrix. The series expansions for these expressions involve fewer terms than previously known results. For the trace, we also present a new algorithm that is linear in the size of the matrix and the degree of truncation, which is optimal.

Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as $k$-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an $k\times k$ Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a $k\times k$ submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this paper, we show that if $R$ tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables. Our proof technique uses the De Bruijn identity and properties of Pfaffians; it also allows to prove that the high powers of these eigenvalues are independent. These results extend to any potential beyond the Gaussian case, as long as radial symmetry holds; this includes for instance truncations of quaternionic unitary matrices, products of quaternionic Ginibre matrices, and the quaternionic spherical ensemble.

We then study the eigenvectors of quaternionic Ginibre matrices. Angles between eigenvectors and the matrix of overlaps both exhibit some specific features that can be compared to the complex case. In particular, we compute the distribution and the limit of the diagonal overlap associated to an eigenvalue that is conditioned to be at the origin. This complements a recent study of overlaps in quaternionic ensembles by Akemann, Förster and Kieburg [1, 2].

In this paper, we study the convergent limits and rates of the eigenvalues and eigenvectors for spiked sample covariance matrices whose spectrum can have multiple bulk components. Our model is an extension of Johnstone’s spiked covariance matrix model. Based on our results, we can extend many statistical applications based on Johnstone’s spiked covariance matrix model.

We consider the empirical eigenvalue distribution of an $m\times m$ principal submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. For $n$ and $m$ large with $\frac{m}{n}=\alpha$, the empirical spectral measure is well approximated by a deterministic measure ${\mu }_{\alpha }$ supported on the unit disc. In earlier work, we showed that for fixed $n$ and $m$, the bounded-Lipschitz distance $d_{\mathrm{\text{BL}}}$ between the empirical spectral measure and the corresponding ${\mu }_{\alpha }$ is typically of order $\sqrt{\frac{log(m)}{m}}$ or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.