Random Matrices: Theory and Applications

Large-dimensional (LD) random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large-dimensional systems. With the fact that we have entered an unprecedented era full of massive amounts of data and large complex systems, RMT is expected to play more important roles in the analysis and design of modern systems. In this paper, we review the key results of RMT and its applications in two emerging fields: wireless communications and deep learning. In wireless communications, we show that RMT can be exploited to design the spectrum sensing algorithms for cognitive radio systems and to perform the design and asymptotic analysis for large communication systems. In deep learning, RMT can be utilized to analyze the Hessian, input–output Jacobian and data covariance matrix of the deep neural networks, thereby to understand and improve the convergence and the learning speed of the neural networks. Finally, we highlight some challenges and opportunities in applying RMT to the practical large-dimensional systems.

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of random circulant matrices with independent Brownian motion entries, as the dimension of the matrix tends to $\infty$. Our derivation is based on the trace formula of circulant matrix, method of moments and some combinatorial techniques.

In this paper, we give bounds on the variance of the number of points of the Circular and the Gaussian $\beta$ Ensemble in arcs of the unit circle or intervals of the real line. These bounds are logarithmic with respect to the renormalized length of these sets, which is expected to be optimal up to a multiplicative constant depending only on $\beta$.

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight

It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39(2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto $\sigma$ form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the $\sigma$-form of the Painlevé II equation at the soft-edge limit, respectively.

In addition, for the special case $A=0,B=1$, we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight $w_{\alpha }(x,t)=x^{\alpha }{\mathrm{\text{e}}}^{-x^2-2tx},x\in {ℝ}^{+}$, which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].

We study the limiting distribution of a pair counting statistics of the form ${\sum }_{1\le i\ne j\le N}f(L_N({𝜃}_i-{𝜃}_j))$ for the circular $\beta$-ensemble (C$\beta$E) of random matrices for sufficiently smooth test function $f$ and $L_N=O(N).$ For $\beta =2$ and $L_N=N$ our results are inspired by a classical result of Montgomery on pair correlation of zeros of Riemann zeta function.

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation ${𝔼}_{\phi }\left[{(z-X-f(X)Y{f}^{\ast }(X))}^{-1}|X\right]$ for free random variables $X,Y$ and a Borel function $f$ is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form $X+f(X)Y{f}^{\ast }(X)$. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J. 2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain quadrature property, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.

The Pearcey process is a universal point process in random matrix theory and depends on a parameter $\rho \in ℝ$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound:

We prove the Marchenko–Pastur law for the eigenvalues of $p\times p$ sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the $p$ coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order $d$, i.e. the coordinates of the data are all $\left(\genfrac{}{}{0.0pt}{}{n}{d}\right)$ different products of $d$ variables chosen from a set of $n$ independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is $o(p)$, and for the random tensor model as long as $d=o(n^{1/3})$. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

For linear spectral statistics of Wigner matrix, we establish a CLT when the entries are independent, standardised with however inhomogeneous fourth moments. Formulas for the expectation and the variance of the Gaussian limiting distribution are given. An application to the normalized stochastic block model is further proposed.