Related papers: Extreme singular values of inhomogeneous sparse ra…

We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erd\"os-Renyi random graph with $N$ vertices of average degree $Z$, inserting a real symmetric $d \times d$ random block at each non-vanishing…

Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$…

For a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random…

We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that…

A large i.i.d. random matrix with deterministic low-rank perturbation has been extensively studied, particularly in the aspects of the ESD (Empirical Spectral Distribution) and the outliers of eigenvalues. In this work, we investigate the…

Let $X=(x_{ij})\in\mathbb{R}^{N\times n}$ be a rectangular random matrix with i.i.d. entries (we assume $N/n\to\mathbf{a}>1$), and denote by $\sigma_{min}(X)$ its smallest singular value. When entries have mean zero and unit second moment,…

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary)…

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…

In this work we consider general non-Hermitian square random matrices $X$ that include a wide class of random band matrices with independent entries. Whereas the existence of limiting density is largely unknown for these inhomogeneous…

We study the gaps between consecutive singular values of random rectangular matrices. Specifically, if $M$ is an $n \times p$ random matrix with independent and identically distributed entries and $\Sigma$ is a $n \times n$ deterministic…

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le \Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random matrix…

We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu.…

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower…

We consider extremal eigenvalues of sparse random matrices, a class of random matrices including the adjacency matrices of Erd\H{o}s-R\'{e}nyi graphs $\mathcal{G}(N,p)$. Recently, it was shown that the leading order fluctuations of extremal…

Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this…

We study the lower tail behavior of the least singular value of an $n\times n$ random matrix $M_n := M+N_n$, where $M$ is a fixed complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a random matrix, each of whose entries is…

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…

Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where $n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we prove that the distribution of the maximal component of $\mathbf{X}$ and the…

We consider non-Hermitian random matrices $X \in \mathbb{C}^{n \times n}$ with general decaying correlations between their entries. For large $n$, the empirical spectral distribution is well approximated by a deterministic density,…