1,819 research outputs found
From Random Matrices to Quasiperiodic Jacobi Matrices via Orthogonal Polynomials
We present an informal review of results on asymptotics of orthogonal
polynomials, stressing their spectral aspects and similarity in two cases
considered. They are polynomials orthonormal on a finite union of disjoint
intervals with respect to the Szego weight and polynomials orthonormal on R
with respect to varying weights and having the same union of intervals as the
set of oscillations of asymptotics. In both cases we construct double infinite
Jacobi matrices with generically quasiperiodic coefficients and show that each
of them is an isospectral deformation of another. Related results on asymptotic
eigenvalue distribution of a class of random matrices of large size are also
shortly discussed
Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases
Orthogonal polynomials on the unit circle: New results
We announce numerous new results in the theory of orthogonal polynomials on
the unit circle
The Nevai Condition
We study Nevai's condition that for orthogonal polynomials on the real line,
where is the CD
kernel. We prove that it holds for the Nevai class of a finite gap set
uniformly on the spectrum and we provide an example of a regular measure on
where it fails on an interval
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