A Simple Approach to Global Regime of the Random Matrix Theory

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular Gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensio...

Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product $\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is Hermitian positive semi-definite. Let $\lambda_1,\lambda_2,...,\lambda_N$ be eigenvalues of $\X...

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-c...

Non-asymptotic theory of random matrices strives to investigate the spectral properties of random matrices, which are valid with high probability for matrices of a large fixed size. Results obtained in this framework find their applications in high-dimensional convexity, analysis of convergence of algorithms, as well as in random matrix theory itself. In these notes we survey some recent results in this area and describe the techniques aimed for obtaining explicit probability...

The local eigenvalue statistics of large random matrices near a hard edge transitioning into a soft edge are described by the Bessel process associated with a large parameter $\alpha$. For this point process, we obtain 1) exponential moment asymptotics, up to and including the constant term, 2) asymptotics for the expectation and variance of the counting function, 3) several central limit theorems and 4) a global rigidity upper bound.

Let $ X_{n} $ be $ n\times N $ random complex matrices, $R_{n}$ and $T_{n}$ be non-random complex matrices with dimensions $n\times N$ and $n\times n$, respectively. We assume that the entries of $ X_{n} $ are independent and identically distributed, $ T_{n} $ are nonnegative definite Hermitian matrices and $T_{n}R_{n}R_{n}^{*}= R_{n}R_{n}^{*}T_{n} $. The general information-plus-noise type matrices are defined by $C_{n}=\frac{1}{N}T_{n}^{\frac{1}{2}} \left( R_{n} +X_{n}\ri...

This paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions (a.e.d.'s) of sums and products of large random matrices. We show that the result can be derived directly as a consequence of two common assumptions, and matches the results obtained from using R- and S-transforms in free probability theory. We also give a direct derivation of the a.e.d. of the sum of certain random matrices which are not free. This is used to determine the ...

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_{n}$ and $B_{n}$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_{n}$ (i.e. $A_{n}+U_{n}^{\ast}B_{n}U_{n}$) is studied in the limit of large matrix order $n$. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in term...

We compute analytically the distribution and moments of the largest eigenvalues/singular values and resolvent statistics for random matrices with (i) non-negative entries, (ii) small rank, and (iii) prescribed sums of rows/columns. Applications are discussed in the context of Mean First Passage Time of random walkers on networks, and the calculation of network "influence" metrics. The analytical results are corroborated by numerical simulations.

We consider $n^2\times n^2$ real symmetric and hermitian matrices $M_n$, which are equal to sum of $m_n$ tensor products of vectors $X^\mu=B(Y^\mu\otimes Y^\mu)$, $\mu=1,\dots,m_n$, where $Y^\mu$ are i.i.d. random vectors from $\mathbb R^n (\mathbb C^n)$ with zero mean and unit variance of components, and $B$ is an $n^2\times n^2$ positive definite non-random matrix. We prove that if $m_n/n^2\to c\in [0,+\infty)$ and the Normalized Counting Measure of eigenvalues of $BJB$, wh...