arXiv subjects
Publications and source records attributed to A. Khorunzhy.
We study the spectral norm of N-dimensional hermitian random matrices whose entries are zero outside of the band of the width b along the principal diagonal. Inside this band the elements are given by gaussian centered jointly independent random variables with the variance of the order 1/b. We show that if b tends to infinity faster than the third power of log N, then the spectral norm is bounded with probability 1.
We consider the family of N-dimensional real symmetric matrices H with random independent entries whose variance is determined by a function U((x-y)/b). In the limit of (relatively) narrow band width 1<<b<<N, we obtain explicitly first terms of 1/b-expansion of the resolvent of H. The expressions derived show that the rate of decay of U(t) determines several scale of the universal form of the eigenvalue correlation function. In particular, in the case of U(t) = o(1/t^3), the Altshuller-Shklovski asymptotics with the ratio N/b^2 is obtained.
We generalise the Erdos-Renyi limit theorem on the maximum of the partial sums of random variables to the case when the number of terms in these sums is randomly distributed. Certain relations between the limiting theorems of this type and the spectral theory of random graphs and random matrices are discussed.
We describe one interpretation of the q-Catalan numbers in frameworks of random matrix theory and weighted partitions of the set of integers.
We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency matrix of G. These relations allow one to study the corresponding eigenvalue distribution function; we show that its density has an infinite support in contrast to the case of the ordinary discrete Laplacian.
We describe the resolvent approach for the rigorous study of the mescoscopic regime of Hermitian matrix spectra. We present results reflecting the universal behavior of the smoothed density of eigenvalue distribution of large random matrices
We consider the ensemble of N-dimensional random symmetric matrices A that have, in average, p non-zero elements per row. We study the asymptotic behavior of the norm of A in the limit of infinitely increasing N and p. We prove that the value p= log N is the critical one for the norm to be either bounded or not. The arguments are based on the calculus of the tree-type graphs. Asymptotic properties of sparse random matrices essentially depend on the typical degree of a tree vertex that we show to be finite.