On the spectrum of sum and product of non-hermitian random matrices

In this short note, we revisit the work of T. Tao and V. Vu on large non-hermitian random matrices with independent and identically distributed entries with mean zero and unit variance. We prove under weaker assumptions that the limit spectral distribution of sum and product of non-hermitian random matrices is universal. As a byproduct, we show that the generalized eigenvalues distribution of two independent matrices converges almost surely to the uniform measure on the Riemann sphere.Comment: 8 pages, statement of main theorem slightly improve

Exact Separation of Eigenvalues of Large Dimensional Noncentral Sample Covariance Matrices

Let \bbB_n =\frac{1}{n}(\bbR_n + \bbT^{1/2}_n \bbX_n)(\bbR_n + \bbT^{1/2}_n \bbX_n)^* where \bbX_n is a $p \times n$ matrix with independent standardized random variables, \bbR_n is a $p \times n$ non-random matrix, representing the information, and \bbT_{n} is a $p \times p$ non-random nonnegative definite Hermitian matrix. Under some conditions on \bbR_n \bbR_n^* and \bbT_n , it has been proved that for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all $p$ sufficiently large. The purpose of this paper is to carry on with the study of the support of the limit spectral distribution, and we show that there is an exact separation phenomenon: with probability one, the proper number of eigenvalues lie on either side of these intervals

Limit spectral distribution for non-degenerate hypersurface singularities

We establish Kyoji Saito’s continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito’s notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp

Enumeration of chord diagrams on many intervals and their non-orientable analogs

Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio

Local Single Ring Theorem

The Single Ring Theorem, by Guionnet, Krishnapur and Zeitouni, describes the empirical eigenvalue distribution of a large generic matrix with prescribed singular values, i.e. an $N\times N$ matrix of the form $A=UTV$, with $U, V$ some independent Haar-distributed unitary matrices and $T$ a deterministic matrix whose singular values are the ones prescribed. In this text, we give a local version of this result, proving that it remains true at the microscopic scale $(\log N)^{-1/4}$. On our way to prove it, we prove a matrix subordination result for singular values of sums of non Hermitian matrices, as Kargin did for Hermitian matrices. This allows to prove a local law for the singular values of the sum of two non Hermitian matrices and a delocalization result for singular vectors.Comment: 33 pages, 2 figures. In version v2: hypothesis of the main theorem slightly weakened, proof adapted. In version v4: some of the proofs simplified, some of the appendix statements fixed, Remarks added, typos correcte

Perturbations of diagonal matrices by band random matrices

We exhibit an explicit formula for the spectral density of a (large) random matrix which is a diagonal matrix whose spectral density converges, perturbated by the addition of a symmetric matrix with Gaussian entries and a given (small) limiting variance profile.Comment: 7 pages, 4 figure