39 research outputs found
Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight
We study asymptotics of the recurrence coefficients of orthogonal polynomials
associated to the generalized Jacobi weight, which is a weight function with a
finite number of algebraic singularities on . The recurrence
coefficients can be written in terms of the solution of the corresponding
Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent
method of Deift and Zhou, we analyze the Riemann-Hilbert problem, and obtain
complete asymptotic expansions of the recurrence coefficients. We will
determine explicitly the order terms in the expansions. A critical step
in the analysis of the Riemann-Hilbert problem will be the local analysis
around the algebraic singularities, for which we use Bessel functions of
appropriate order.Comment: 31 pages, 6 figures, 21 reference
Universality for eigenvalue correlations at the origin of the spectrum
We establish universality of local eigenvalue correlations in unitary random
matrix ensembles (1/Z_n) |\det M|^{2\alpha} e^{-n\tr V(M)} dM near the origin
of the spectrum. If V is even, and if the recurrence coefficients of the
orthogonal polynomials associated with |x|^{2\alpha} e^{-nV(x)} have a regular
limiting behavior, then it is known from work of Akemann et al., and Kanzieper
and Freilikher that the local eigenvalue correlations have universal behavior
described in terms of Bessel functions. We extend this to a much wider class of
confining potentials V. Our approach is based on the steepest descent method of
Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This
method was used by Deift et al. to establish universality in the bulk of the
spectrum. A main part of the present work is devoted to the analysis of a local
Riemann-Hilbert problem near the origin.Comment: 28 pages, 6 figures, technical problem in second version removed, to
appear in Commun. Math. Phy
The RiemannâHilbert approach to strong asymptotics for orthogonal polynomials on [â1,1]
We consider polynomials that are orthogonal on [â1,1] with respect to a modified Jacobi weight (1â ) (1+ ) ( ), with , >â1 and real analytic and strictly positive on [â1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [â1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [â1,1]. For the asymptotic analysis we use the steepest descent technique for RiemannâHilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the SzegĆ function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions
Universality of a double scaling limit near singular edge points in random matrix models
We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr
V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining
potential V_{s,t} is such that the limiting mean density of eigenvalues (as
n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of
its support. The main purpose of this paper is to prove universality of the
eigenvalue correlation kernel in a double scaling limit. The limiting kernel is
built out of functions associated with a special solution of the P_I^2
equation, which is a fourth order analogue of the Painleve I equation. In order
to prove our result, we use the well-known connection between the eigenvalue
correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal
polynomials, together with the Deift/Zhou steepest descent method to analyze
the RH problem asymptotically. The key step in the asymptotic analysis will be
the construction of a parametrix near the singular endpoint, for which we use
the model RH problem for the special solution of the P_I^2 equation.
In addition, the RH method allows us to determine the asymptotics (in a
double scaling limit) of the recurrence coefficients of the orthogonal
polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}.
The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the
asymptotics.Comment: 32 pages, 3 figure
Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case
We consider the sample covariance matrices of large data matrices which have
i.i.d. complex matrix entries and which are non-square in the sense that the
difference between the number of rows and the number of columns tends to
infinity. We show that the second-order correlation function of the
characteristic polynomial of the sample covariance matrix is asymptotically
given by the sine kernel in the bulk of the spectrum and by the Airy kernel at
the edge of the spectrum. Similar results are given for real sample covariance
matrices
Universality for orthogonal and symplectic Laguerre-type ensembles
We give a proof of the Universality Conjecture for orthogonal (beta=1) and
symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the
spectrum as well as at the hard and soft spectral edges. Our results are stated
precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5,
1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation
and cluster functions, gap probabilities and the distributions of the largest
and smallest eigenvalues. Corresponding results for unitary (beta=2)
Laguerre-type ensembles have been proved by the fourth author in [23]. The
varying weight case at the hard spectral edge was analyzed in [13] for beta=2:
In this paper we do not consider varying weights.
Our proof follows closely the work of the first two authors who showed in
[7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use
the version of the orthogonal polynomial method presented in [25], [22] to
analyze the local eigenvalue statistics. The necessary asymptotic information
on the Laguerre-type orthogonal polynomials is taken from [23].Comment: 75 page
The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation
We establish the existence of a real solution y(x,T) with no poles on the
real line of the following fourth order analogue of the Painleve I equation,
x=Ty-({1/6}y^3+{1/24}(y_x^2+2yy_{xx})+{1/240}y_{xxxx}). This proves the
existence part of a conjecture posed by Dubrovin. We obtain our result by
proving the solvability of an associated Riemann-Hilbert problem through the
approach of a vanishing lemma. In addition, by applying the Deift/Zhou
steepest-descent method to this Riemann-Hilbert problem, we obtain the
asymptotics for y(x,T) as x\to\pm\infty.Comment: 27 pages, 5 figure