Random Matrices: Theory and Applications

A number of random matrix ensembles permitting exact determination of their eigenvalue and eigenvector statistics maintain this property under a rank $1$ perturbation. Considered in this review are the additive rank $1$ perturbation of the Hermitian Gaussian ensembles, the multiplicative rank $1$ perturbation of the Wishart ensembles, and rank $1$ perturbations of Hermitian and unitary matrices giving rise to a two-dimensional support for the eigenvalues. The focus throughout is on exact formulas, which are typically the result of various integrable structures. The simplest is that of a determinantal point process, with others relating to partial differential equations implied by a formulation in terms of certain random tridiagonal matrices. Attention is also given to eigenvector overlaps in the setting of a rank $1$ perturbation.

Motivated by the stochastic block model, we investigate a class of Wigner-type matrices with certain block structures and establish a CLT for the corresponding linear spectral statistics (LSS) via the large-deviation bounds from local law and the cumulant expansion formula. We apply the results to the stochastic block model. Specifically, a class of renormalized adjacency matrices will be block-Wigner-type matrices. Further, we show that for certain estimator of such renormalized adjacency matrices, which will be no longer Wigner-type but share long-range non-decaying weak correlations among the entries, the LSS of such estimators will still share the same limiting behavior as those of the block-Wigner-type matrices, thus enabling hypothesis testing about stochastic block model.

Motivated by the general matrix deviation inequality for i.i.d. ensemble Gaussian matrix [R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, 2018), doi:10.1017/9781108231596 of Theorem 11.1.5], we show that this property holds for the ${\ell }_p$-norm with $1\le p<\infty$ and i.i.d. ensemble sub-Gaussian matrices, i.e. random matrices with i.i.d. mean-zero, unit variance, sub-Gaussian entries. As a consequence of our result, we establish the Johnson–Lindenstrauss lemma from ${\ell }_2^n$-space to ${\ell }_p^m$-space for all i.i.d. ensemble sub-Gaussian matrices.

The stochastic block model (SBM) is an extension of the Erdős–Rényi graph and has applications in numerous fields, such as data analysis, recovering community structure in graph data and social networks. In this paper, we consider the normal central SBM adjacency matrix with $K$ communities of arbitrary sizes. We derive an explicit formula for the limiting empirical spectral density function when the size of the matrix tends to infinity. We also obtain an upper bound for the operator norm of such random matrices by means of the Stieltjes transform and random matrix theory.

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.

In this paper, we generalize the formula for the fourth moment of a random determinant to account for entries with asymmetric distribution. We also derive the second moment of a random Gram determinant.

One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+𝜃v{v}^T$ with $A={(a_{ij})}_{1\le i,j\le n}\in {ℝ}^{n\times n}$ symmetric, ${(a_{ij})}_{1\le i\le j\le n}$ i.i.d. with centered standard normal distributions, and $𝜃>0,v\in {𝕊}^{n-1}.$ It is well known ${\lambda }_1(P),$ the largest eigenvalue of $P,$ has a phase transition at ${𝜃}_0=1$: when $𝜃\le 1,$ ${\lambda }_1(P)\underset{}{\overset{\mathrm{\text{a.s.}}}{\to }}2,$ whereas for $𝜃>1,$ ${\lambda }_1(P)\underset{}{\overset{\mathrm{\text{a.s.}}}{\to }}𝜃+{𝜃}^{-1}.$ Under more general conditions, the limiting behavior of ${\lambda }_1(P),$ appropriately normalized, has also been established: it is normal if $\left|\right|v|{|}_{\infty }=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $\alpha \in (0,4),$ the fluctuations are shown to be universal and dependent on $𝜃$ but not on $v,$ whereas a subfamily of the edge case $\alpha =4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at ${𝜃}_0=1,{𝜃}_0\in [1,\frac{128}{89}],$ respectively. These results build on our previous work which analyzes the asymptotic behavior of ${\lambda }_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.

Suppose that $y_{\alpha }^{(i)},i=1,\dots ,k,\alpha =1,\dots ,m$ are i.i.d. copies of random vector $y={(y_1,y_2,\dots ,y_p)}^T$. Let