arXiv subjects

Victor Pérez-Abreu

Publications and source records attributed to Victor Pérez-Abreu.

8 recordsLinked to original sources

arXiv2024

Extreme eigenvalues of Laplacian random matrices with Gaussian entries

A Laplacian matrix is a real symmetric matrix whose row and column sums are zero. We investigate the limiting distribution of the largest eigenvalues of a Laplacian random matrix with Gaussian entries. Unlike many classical matrix ensembles, this random matrix model contains dependent entries. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics. In particular, after properly shifting and scaling, we show that the largest eigenvalue converges to the Gumbel distribution as the dimension of the matrix tends to infinity. While the largest diagonal entry is also shown to have Gumbel fluctuations, there is a rather surprising difference between its deterministic centering term and the centering term required for the largest eigenvalues.

math.PR↗

arXiv2021

Extremal Laws for Laplacian Random Matrices

For an $n\times n$ Laplacian random matrix $L$ with Gaussian entries it is proven that the fluctuations of the largest eigenvalue and the largest diagonal entry of $L/\sqrt{n-1}$ are Gumbel. We first establish suitable non-asymptotic estimates and bounds for the largest eigenvalue of $L$ in terms of the largest diagonal element of $L$. An expository review of existing results for the asymptotic spectrum of a Laplacian random matrix is also presented, with the goal of noting the differences from the corresponding classical results for Wigner random matrices. Extensions to Laplacian block random matrices are indicated.

math.PR↗

arXiv2016

On the Free Fractional Wishart Process

We investigate the process of eigenvalues of a fractional Wishart process defined as N=B*B, where B is a matrix fractional Brownian motion recently studied by Nualart and Pérez-Abreu. Using stochastic calculus with respect to the Young integral we show that the eigenvalues do not collide at any time with probability one. When the matrix process B has entries given by independent fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ we derive a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding fractional Wishart process. Finally a functional limit theorem for the empirical measure-valued process of eigenvalues of a fractional Wishart process is obtained. The limit is characterized and referred to as the free fractional Wishart process which constitutes the family of fractional dilations of the free Poisson distribution.

math.PR↗

arXiv2016

On the Universality of the Non-singularity of General Ginibre and Wigner Random Matrices

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.

math.PR↗

arXiv2015

On the process of the eigenvalues of a Hermitian Lévy process

The dynamics of the eigenvalues (semimartingales) of a Lévy process $X$ with values in Hermitian matrices is described in terms of Itô stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of $X$ is also studied. If $X$ has a jump at time $t$ two different situations are considered, depending on the commutativity of $X(t)$ and $X(t-)$. In the commutative case all the eigenvalues jump at time $t$ only when the jump of $X$ is of full rank. In the noncommutative case, $X$ jumps at time $t$ if and only if all the eigenvalues jump at that time when the jump of $X$ is of rank one.

math.PR↗

arXiv2015

A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion

A functional limit theorem for the empirical measure-valued process of eigenvalues of a matrix fractional Brownian motion is obtained. It is shown that the limiting measure-valued process is the non-commutative fractional Brownian motion recently introduced by Nourdin and Taqqu. Young and Skorohod stochastic integral techniques and fractional calculus are the main tools used.

math.PR↗

arXiv2014

On the eigenvalue process of a matrix fractional Brownian motion

We investigate the process of eigenvalues of a symmetric matrix-valued process which upper diagonal entries are independent one-dimensional Hölder continuous Gaussian processes of order gamma in (1/2,1). Using the stochastic calculus with respect to the Young's integral we show that these eigenvalues do not collide at any time with probability one. When the matrix process has entries that are fractional Brownian motions with Hurst parameter H in (1/2,1), we find a stochastic differential equation in a Malliavin calculus sense for the eigenvalues of the corresponding matrix fractional Brownian motion. A new generalized version of the Itô formula for the multidimensional fractional Brownian motion is first established.

math.PR↗

arXiv2012

A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions

Classes of multivariate and cone valued infinitely divisible Gamma distributions are introduced. Particular emphasis is put on the cone-valued case, due to the relevance of infinitely divisible distributions on the positive semi-definite matrices in applications. The cone-valued class of generalised Gamma convolutions is studied. In particular, a characterisation in terms of an Itô-Wiener integral with respect to an infinitely divisible random measure associated to the jumps of a Lévy process is established. A new example of an infinitely divisible positive definite Gamma random matrix is introduced. It has properties which make it appealing for modelling under an infinite divisibility framework. An interesting relation of the moments of the Lévy measure and the Wishart distribution is highlighted which we suppose to be important when considering the limiting distribution of the eigenvalues .

math.PR↗