arXiv subjects

Jose A. Ramirez

Publications and source records attributed to Jose A. Ramirez.

6 recordsLinked to original sources

arXiv2015

Spiking the random matrix hard edge

We characterize the limiting smallest eigenvalue distributions (or hard edge laws) for sample covariance type matrices drawn from a spiked population. In the case of a single spike, the results are valid in the context of the general beta ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion (beta=1, 2, or 4) ensembles. The limit laws are described in terms of a random integral operators, and partial differential equations satisfied by the corresponding distribution functions are derived as corollaries. We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy-Widom laws). The latter were first described at beta=2 by Baik, Ben Arous, and Peche, and from a unified beta random operator point of view by Bloemendal and Virag.

math.PR↗

arXiv2011

Hard edge tail asymptotics

Let $Λ$ be the limiting smallest eigenvalue in the general (β, a)-Laguerre ensemble of random matrix theory. Here β>0, a >-1; for β=1,2,4 and integer a, this object governs the singular values of certain rank n Gaussian matrices. We prove that P(Λ> λ) = e^{- (β/2) λ+ 2 γλ^{1/2}} λ^{- (γ(γ+1))/(2β) + γ/4} E (β, a) (1+o(1)) as λgoes to infinity, in which γ= (β/2) (a+1)-1 and E(β, a) is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of βand a.

math.PR↗

arXiv2010

Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+β}$ for some positive $β>0$. We prove that the local eigenvalue statistics follows the universal Dyson sine kernel.

math-ph↗

arXiv2009

Bulk Universality for Wigner Matrices

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

math-ph↗

arXiv2008

Diffusion at the random matrix hard edge

We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures (the "beta ensembles") are described by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the so-called hard edge of random matrix theory and sits in complement to the recent work of the authors and B. Virag on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta.

math.PR↗

arXiv2006

Shape of the ground state energy density of Hill's equation with nice Gaussian potential

Consider Hill's operator Q = -D^2 + q(x) in which the potential q(x) is an almost surely continuous and rotation invariant Gaussian process on the circle of perimeter one. Viewing the classical Riccati map as a change of measure, we establish functional integral formulas for the probability density function of the ground state energy and also determine the density's shape.

math.PR↗