Related papers: CLT for non-Hermitian random band matrices with va…
We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by $x,y \in (\bZ/L\bZ)^d$, are independent, centred random variables with variances $s_{xy} = \E…
This paper considers testing the covariance matrices structure based on Wald's score test in large dimensional setting. The hypothesis $H_0: \Sigma =\Sigma_0 $ for a given matrix $\Sigma_0$, which covers the identity hypothesis test and…
We consider sample covariance matrices $S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}$ where $X_N$ is a $N \times p$ real or complex matrix with i.i.d. entries with finite $12^{\rm th}$ moment and $\Sigma_N$ is a $N \times N$…
A method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices. The topological classification of Feynman diagrams, which was actively used in last years for matrix…
We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical…
We consider the spectral properties of a class of regularized estimators of (large) empirical covariance matrices corresponding to stationary (but not necessarily Gaussian) sequences, obtained by banding. We prove a law of large numbers…
We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma_k^{1/2}Z_k$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and $\Sigma_k$ are…
We consider the eigenvalues of a fixed, non-normal matrix subject to a small additive perturbation. In particular, we consider the case when the fixed matrix is a banded Toeplitz matrix, where the bandwidth is allowed to grow slowly with…
We demonstrate that the normalised localization length $\beta$ of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law $\beta=x^*/(1+x^*)$. The scaling parameter of the model is defined as…
We consider Hermitian random band matrices $H$ in $d \geq 1 $ dimensions. The matrix elements $H_{xy},$ indexed by $x, y \in \Lambda \subset \mathbb{Z}^d,$ are independent, uniformly distributed random variable if $|x-y| $ is less than the…
The growth of large-scale structure, together with the geometrical information of cosmic expansion history and cosmological distances, can be used to obtain constraints on the spatial curvature of the universe that probes the early universe…
We describe the distribution of the first finite number of eigenvalues in a newly-forming band of the spectrum of the random Hermitean matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding…
Recently, Kalkreuter obtained the complete Dirac spectrum for an $SU(2)$ lattice gauge theory with dynamical staggered fermions on a $12^4$ lattice for $\beta =1.8$ and $\beta=2.8$. We performed a statistical analysis of his data and found…
We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the…
We prove that under the Brownian evolution on large non-Hermitian matrices the log-determinant converges in distribution to a 2+1 dimensional Gaussian field in the Edwards-Wilkinson regularity class, namely it is logarithmically correlated…
We generalize classical results in spectral graph theory and linear algebra more broadly, from the case where the underlying matrix is Hermitian to the case where it is non-Hermitian. New admissibility conditions are introduced to replace…
We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…
We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is…
We apply the Wiener Hermite (WH) expansion to the non-linear evolution of Large-Scale Structure, and obtain an approximate expression for the matter power spectrum in full order of the expansion. This method allows us to expand any random…
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) =…