Related papers: A Local Limit Theorem and Delocalization of Eigenv…

Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix $n^{-1}U^*U$ and its expectation where $U$ is…

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values…

We study a random band matrix $H=(H_{xy})_{x,y}$ of dimension $N\times N$ with mean-zero complex Gaussian entries, where $x,y$ belong to the discrete torus $(\mathbb{Z}/\sqrt{N}\mathbb{Z})^{2}$. The variance profile…

We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large…

We consider complex sample covariance matrices $M_N=\frac{1}{N}YY^*$ where $Y$ is a $N \times p$ random matrix with i.i.d. entries $Y_{ij}, 1\leq i\leq N, 1\leq j \leq p$ with distribution $F$. Under some regularity and decay assumption on…

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and R\'effy identified the limiting spectral measure…

The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix…

We give a construction which produces irreducible complex rigid local systems on $\Bbb{P}_{\Bbb{C}}^1-\{p_1,\dots,p_s\}$ via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices…

We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions. The main novel…

In this paper, we study the eigenvalues of the GCD matrix $(S_n)$ and the LCM matrix $[S_n]$ defined on $S_n=\{1,2,\ldots,n\}$. We present upper and lower bounds for the smallest and the largest eigenvalues of $(S_n)$ and $[S_n]$ in terms…

We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions…

We consider some random band matrices with band-width $N^\mu$ whose entries are independent random variables with distribution tail in $x^{-\alpha}$. We consider the largest eigenvalues and the associated eigenvectors and prove the…

We consider random $d$-regular graphs on $N$ vertices, with degree $d$ at least $(\log N)^4$. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by…

Let $f=(f_1,\ldots,f_n)$ be a system of $n$ complex homogeneous polynomials in $n$ variables of degree $d$. We call $\lambda\in\mathbb{C}$ an eigenvalue of $f$ if there exists $v\in\mathbb{C}^n\backslash\{0\}$ with $f(v)=\lambda v$,…

Let $A_n$ be an $n \times n$ deterministic matrix and $\Sigma_n$ be a deterministic non-negative matrix such that $A_n$ and $\Sigma_n$ converge in $*$-moments to operators $a$ and $\Sigma$ respectively in some $W^*$-probability space. We…

Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law,…

We study large $N\times N$ power-law random band matrices $H=(H_{ij})$ with centered complex Gaussian entries, where the variances satisfy a power-law decay $\mathbb{E}|H_{ij}|^2\propto (|i-j|/W+1)^{-1-\alpha}$, for some exponent…

We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable…

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset \R^2$) and a variable-exponent growth in…

Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be…