It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.
For matrices drawn from the standard Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE), we prove that the normalized zero counting measure of the permanental characteristic polynomial $Per(zI_N-H_N)$ converges almost surely to the standard Wigner semicircle law on $[-2,2]$, rotated by $\pi/2$ onto...
We study the eigenvalues of the random Laplacian matrix $D - A$, where $A$ is a Wigner matrix with sub-Gaussian entries and the diagonal matrix $D$ contains the row sums of $A$. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering an...
We state and prove a version of Szeg\H{o}'s first limit theorem for multi-Toeplitz operators acting in the full Fock space in $d\geq 2$ letters. In particular we show that the eigenvalue distributions of truncated multi-Toeplitz operators converge to a limiting distribution. We compute this limiting distribution and it...
We establish nonasymptotic spectral comparison results for matrices whose entries are non-linear functions of an iid random field. Under an exponential decay assumption on an $\mathbb L^\infty$-coupling coefficient, we derive high-probability bounds for the Hausdorff distance between their spectra and those of Gaussian...
Marwa Banna, Issa-Mbenard Dabo, F. Merlevède· 0 citations
We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral dis...
Anastasis Kafetzopoulos· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
