880 research outputs found
Sine kernel asymptotics for a class of singular measures
We construct a family of measures on \bbR that are purely singular with
respect to Lebesgue measure, and yet exhibit universal sine-kernel asymptotics
in the bulk. The measures are best described via their Jacobi recursion
coefficients: these are sparse perturbations of the recursion coefficients
corresponding to Chebyshev polynomials of the second kind. We prove convergence
of the renormalized Christoffel-Darboux kernel to the sine kernel for any
sufficiently sparse decaying perturbation
Asymptotics for a determinant with a confluent hypergeometric kernel
We obtain "large gap" asymptotics for a Fredholm determinant with a confluent
hypergeometric kernel. We also obtain asymptotics for determinants with two
types of Bessel kernels which appeared in random matrix theory.Comment: 34 pages, 2 figure
An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles
We survey the current status of universality limits for -point correlation
functions in the bulk and at the edge for unitary ensembles, primarily when the
limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider
underlying measures on compact intervals, and fixed and varying exponential
weights, as well as universality limits for a variety of orthogonal systems.
The scope of the survey is quite narrow: we do not consider ensembles
for , nor general Hermitian matrices with independent entries,
let alone more general settings. We include some open problems
Energy correlations for a random matrix model of disordered bosons
Linearizing the Heisenberg equations of motion around the ground state of an
interacting quantum many-body system, one gets a time-evolution generator in
the positive cone of a real symplectic Lie algebra. The presence of disorder in
the physical system determines a probability measure with support on this cone.
The present paper analyzes a discrete family of such measures of exponential
type, and does so in an attempt to capture, by a simple random matrix model,
some generic statistical features of the characteristic frequencies of
disordered bosonic quasi-particle systems. The level correlation functions of
the said measures are shown to be those of a determinantal process, and the
kernel of the process is expressed as a sum of bi-orthogonal polynomials. While
the correlations in the bulk scaling limit are in accord with sine-kernel or
GUE universality, at the low-frequency end of the spectrum an unusual type of
scaling behavior is found.Comment: 20 pages, 3 figures, references adde
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