656 research outputs found
On the existence of finite critical trajectories in a family of quadratic differentials
In this note, we discuss the possible existence of finite critical
trajectories connecting two zeros a(t) and b(t) of a family of quadratic
differentials satisfying some properties. We treat the cases of holomorphic and
meromorphic quadratic differentials, and we give new proofs concerning the
supports of limit measures of the root-counting measures of the generalized
Laguerre and Jacobi polynomials with varying parameters.Comment: 12 page
The impact of Stieltjes' work on continued fractions and orthogonal polynomials
Stieltjes' work on continued fractions and the orthogonal polynomials related
to continued fraction expansions is summarized and an attempt is made to
describe the influence of Stieltjes' ideas and work in research done after his
death, with an emphasis on the theory of orthogonal polynomials
Remez-type inequalities and their applications
AbstractThe Remez inequality gives a sharp uniform bound on [â1, 1] for real algebraic polynomials p of degree at most n if the Lebesgue measure of the subset of [â1, 1], where |;p|; is at most 1, is known. Remez-type inequalities give bounds for classes of functions on a line segment, on a curve or on a region of the complex plane, given that the modulus of the functions is bounded by 1 on some subset of prescribed measure. This paper offers a survey of the extensive recent research on Remez-type inequalities for polynomials, generalized nonnegative polynomials, exponentials of logarithmic potentials and MĂŒntz polynomials. Remez-type inequalities play a central role in proving other important inequalities for the above classes. The paper illustrates the power of Remez-type inequalities by giving a number of applications
The two periodic Aztec diamond and matrix valued orthogonal polynomials
We analyze domino tilings of the two-periodic Aztec diamond by means of
matrix valued orthogonal polynomials that we obtain from a reformulation of the
Aztec diamond as a non-intersecting path model with periodic transition
matrices. In a more general framework we express the correlation kernel for the
underlying determinantal point process as a double contour integral that
contains the reproducing kernel of matrix valued orthogonal polynomials. We use
the Riemann-Hilbert problem to simplify this formula for the case of the
two-periodic Aztec diamond.
In the large size limit we recover the three phases of the model known as
solid, liquid and gas. We describe fine asymptotics for the gas phase and at
the cusp points of the liquid-gas boundary, thereby complementing and extending
results of Chhita and Johansson.Comment: 80 pages, 20 figures; This is an extended version of the paper that
is accepted for publication in the Journal of the EM
Perturbations of Orthogonal Polynomials With Periodic Recursion Coefficients
We extend the results of Denisov-Rakhmanov, Szego-Shohat-Nevai, and
Killip-Simon from asymptotically constant orthogonal polynomials on the real
line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC.
The key tool is a characterization of the isospectral torus that is well
adapted to the study of perturbations.Comment: 64 pages, to appear in Ann. of Mat
Random matrix minor processes related to percolation theory
This paper studies a number of matrix models of size n and the associated
Markov chains for the eigenvalues of the models for consecutive n's. They are
consecutive principal minors for two of the models, GUE with external source
and the multiple Laguerre matrix model, and merely properly defined consecutive
matrices for the third one, the Jacobi-Pineiro model; nevertheless the
eigenvalues of the consecutive models all interlace. We show: (i) For each of
those finite models, we give the transition probability of the associated
Markov chain and the joint distribution of the entire interlacing set of
eigenvalues; we show this is a determinantal point process whose extended
kernels share many common features. (ii) To each of these models and their set
of eigenvalues, we associate a last-passage percolation model, either finite
percolation or percolation along an infinite strip of finite width, yielding a
precise relationship between the last passage times and the eigenvalues. (iii)
Finally it is shown that for appropriate choices of exponential distribution on
the percolation, with very small means, the rescaled last passage times lead to
the Pearcey process; this should connect the Pearcey statistics with random
directed polymers.Comment: 57 pages, 15 figures; more discussion on the relation to percolation
and directed polymer; some more references adde
Fast, reliable and unrestricted iterative computation of Gauss-Hermite and Gauss-Laguerre quadratures
Methods for the computation of classical Gaussian quadrature rules are described which are effective both for small and large degree. These methods are reliable because the iterative computation of the nodes has guaranteed convergence, and they are fast due to their fourth-order convergence and its asymptotic exactness for an appropriate selection of the variables. For Gauss?Hermite and Gauss?Laguerre quadratures, local Taylor series can be used for computing efficiently the orthogonal polynomials involved, with exact initial values for the Hermite case and first values computed with a continued fraction for the Laguerre case. The resulting algorithms have almost unrestricted validity with respect to the parameters. Full relative precision is reached for the Hermite nodes, without any accuracy loss and for any degree, and a mild accuracy loss occurs for the Hermite and Laguerre weights as well as for the Laguerre nodes. These fast methods are exclusively based on convergent processes, which, together with the high order of convergence of the underlying iterative method, makes them particularly useful for high accuracy computations. We show examples of very high accuracy computations (of up to 1000 digits of accuracy)
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