27 research outputs found
Strong Asymptotics of Hermite-Pad\'e Approximants for Angelesco Systems
In this work type II Hermite-Pad\'e approximants for a vector of Cauchy
transforms of smooth Jacobi-type densities are considered. It is assumed that
densities are supported on mutually disjoint intervals (an Angelesco system
with complex weights). The formulae of strong asymptotics are derived for any
ray sequence of multi-indices.Comment: 40 page
Symmetric Contours and Convergent Interpolation
The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as
applied to the multipoint Pad\'e approximants is the fact that given a germ of
an algebraic function and a sequence of rational interpolants with free poles
of the germ, if there exists a contour that is "symmetric" with respect to the
interpolation scheme, does not separate the plane, and in the complement of
which the germ has a single-valued continuation with non-identically zero jump
across the contour, then the interpolants converge to that continuation in
logarithmic capacity in the complement of the contour. The existence of such a
contour is not guaranteed. In this work we do construct a class of pairs
interpolation scheme/symmetric contour with the help of hyperelliptic Riemann
surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author.
We consider rational interpolants with free poles of Cauchy transforms of
non-vanishing complex densities on such contours under mild smoothness
assumptions on the density. We utilize -extension of the
Riemann-Hilbert technique to obtain formulae of strong asymptotics for the
error of interpolation
Nuttall's theorem with analytic weights on algebraic S-contours
Given a function holomorphic at infinity, the -th diagonal Pad\'e
approximant to , denoted by , is a rational function of type
that has the highest order of contact with at infinity. Nuttall's
theorem provides an asymptotic formula for the error of approximation
in the case where is the Cauchy integral of a smooth density
with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's
theorem is extended to Cauchy integrals of analytic densities on the so-called
algebraic S-contours (in the sense of Nuttall and Stahl)
Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary
We investigate a two-dimensional statistical model of N charged particles
interacting via logarithmic repulsion in the presence of an oppositely charged
compact region K whose charge density is determined by its equilibrium
potential at an inverse temperature corresponding to \beta = 2. When the charge
on the region, s, is greater than N, the particles accumulate in a neighborhood
of the boundary of K, and form a determinantal point process on the complex
plane. We investigate the scaling limit, as N \to \infty, of the associated
kernel in the neighborhood of a point on the boundary under the assumption that
the boundary is sufficiently smooth. We find that the limiting kernel depends
on the limiting value of N/s, and prove universality for these kernels. That
is, we show that, the scaled kernel in a neighborhood of a point \zeta \in
\partial K can be succinctly expressed in terms of the scaled kernel for the
closed unit disk, and the exterior conformal map which carries the complement K
to the complement of the closed unit disk. When N / s \to 0 we recover the
universal kernel discovered by Doron Lubinsky in Universality type limits for
Bergman orthogonal polynomials, Comput. Methods Funct. Theory, 10:135-154,
2010.Comment: 25 pages, 11 figure
The reciprocal Mahler ensembles of random polynomials
We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree N whose Mahler measure is bounded by a constant. After a change of variables, this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval [−2,2] on the real axis in the complex plane. In the complex (real) case, the random roots form a determinantal (Pfaffian) point process, and in both cases, the empirical measure on roots converges weakly to the arcsine distribution supported on [−2,2]. Outside this region, the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from [−2,2]. These kernels as well as the scaling limits for the kernels in the bulk (−2,2) and at the endpoints {−2,2} are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels
An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC
Let {φi}∞i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say En(μ), of random polynomials
Pn(z):=∑i=0nηiφi(z),
where η0,…,ηn are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that En(|dξ|) admits an asymptotic expansion of the form
En(|dξ|)∼2πlog(n+1)+∑p=0∞Ap(n+1)−p
(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case En(μ) admits an analogous expansion with coefficients the Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same)