Random Matrices: Theory and Applications

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for these traces of powers, which imply central limit theorems for polynomial linear eigenvalue statistics. In contrast to the usual situation in random matrix theory, in our approach general, nonnormal matrices turn out to be easier to study than Hermitian matrices.

The aim of this paper is to investigate the Kolmogorov distance of the Circular Law to the empirical spectral distribution of non-Hermitian random matrices with independent entries. The optimal rate of convergence is determined by the Ginibre ensemble and is given by $n^{-1/2}$. A smoothing inequality for complex measures that quantitatively relates the uniform Kolmogorov-like distance to the concentration of logarithmic potentials is shown. Combining it with results from Local Circular Laws, we apply it to prove nearly optimal rate of convergence to the Circular Law in Kolmogorov distance. Furthermore, we show that the same rate of convergence holds for the empirical measure of the roots of Weyl random polynomials.

Determinantal point processes are point processes whose correlation functions are given by determinants of matrices. The entries of these matrices are given by one fixed function of two variables, which is called the kernel of the point process. It is well known that there are different kernels that induce the same correlation functions. We classify all the possible transformations of a kernel that leave the induced correlation functions invariant, restricting to the case of symmetric kernels.

We describe the non-backtracking spectrum of a stochastic block model with connection probabilities $p_{\mathrm{\text{in}}},p_{\mathrm{\text{out}}}=\omega (logn)/n$. In this regime we answer a question posed in [L. Dall’Amico, R. Couillet and N. Tremblay, Revisiting the Bethe–Hessian: Improved community detection in sparse heterogeneous graphs, in Advances in Neural Information Processing Systems (2019), pp. 4039–4049] regarding the existence of a real eigenvalue “inside” the bulk, close to the location $\frac{p_{\mathrm{\text{in}}}+p_{\mathrm{\text{out}}}}{p_{\mathrm{\text{in}}}-p_{\mathrm{\text{out}}}}$. We also introduce a variant of the Bauer–Fike theorem well suited for perturbations of quadratic eigenvalue problems, which could be of independent interest.

It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the $\sigma$-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the $2k$th moment of the limit equals a weighted sum over different types of pair-partitions of the set $\{1,2,\dots ,2k\}$ and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

Let $N_n$ be an $n\times n$ complex random matrix, each of whose entries is an independent copy of a centered complex random variable $z$ with finite nonzero variance ${\sigma }^2$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(\sigma \sqrt{n})$ converges to the uniform distribution on the unit disc in $ℂ$.

A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the random matrix $xI-N_n/(\sigma \sqrt{n})$ (where $x\in ℂ$ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using machinery from additive combinatorics or any net arguments), we show that for any fixed complex matrix $M$ with operator norm at most $n^{3/4-𝜖}$ and for all $\eta \ge 0$,

Our proof relies on the solution to the so-called counting problem in inverse Littlewood–Offord theory, developed by Ferber, Luh, Samotij, and the author, a novel complex anti-concentration inequality, and a “rounding trick” based on controlling the $\infty \to 2$ operator norm of heavy-tailed random matrices.

We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular, for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups, the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular, for symmetric groups of increasing order) families of stochastically independent random convolution operators converge in moments to free circular elements. Finally, in the Gaussian case, we provide Central Limit Theorems for linear eigenvalue statistics.

The spectral density for random matrix $\beta$ ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of $\beta$, which for even $\beta$ is a polynomial of degree $\beta (N-1)$. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree $\beta +1$. This equation, and its companion for the resolvent, are given explicitly for $\beta =2$ and $4$ for all three classical cases, and also for $\beta =6$ in the Gaussian case. Known dualities for the spectral moments relating $\beta$ to $4/\beta$ then imply corresponding differential equations in the case $\beta =1$, and for the Gaussian ensemble, the case $\beta =2/3$. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their $1/N$ expansions, along with first-order differential equations for the coefficients of the $1/N$ expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.