71 research outputs found
The Riemann–Hilbert approach to strong asymptotics for orthogonal polynomials on [−1,1]
We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1− ) (1+ ) ( ), with , >−1 and real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], 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 [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann–Hilbert 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 Szegő 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
Root asymptotics of spectral polynomials for the Lame operator
The study of polynomial solutions to the classical Lam\'e equation in its
algebraic form, or equivalently, of double-periodic solutions of its
Weierstrass form has a long history. Such solutions appear at integer values of
the spectral parameter and their respective eigenvalues serve as the ends of
bands in the boundary value problem for the corresponding Schr\"odinger
equation with finite gap potential given by the Weierstrass -function on
the real line. In this paper we establish several natural (and equivalent)
formulas in terms of hypergeometric and elliptic type integrals for the density
of the appropriately scaled asymptotic distribution of these eigenvalues when
the integer-valued spectral parameter tends to infinity. We also show that this
density satisfies a Heun differential equation with four singularities.Comment: final version, to appear in Commun. Math. Phys.; 13 pages, 3 figures,
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Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
We study a model of non-intersecting squared Bessel processes in the
confluent case: all paths start at time at the same positive value , remain positive, and are conditioned to end at time at . In
the limit , after appropriate rescaling, the paths fill out a
region in the -plane that we describe explicitly. In particular, the paths
initially stay away from the hard edge at , but at a certain critical
time the smallest paths hit the hard edge and from then on are stuck to
it. For we obtain the usual scaling limits from random matrix
theory, namely the sine, Airy, and Bessel kernels. A key fact is that the
positions of the paths at any time constitute a multiple orthogonal
polynomial ensemble, corresponding to a system of two modified Bessel-type
weights. As a consequence, there is a matrix valued
Riemann-Hilbert problem characterizing this model, that we analyze in the large
limit using the Deift-Zhou steepest descent method. There are some novel
ingredients in the Riemann-Hilbert analysis that are of independent interest.Comment: 59 pages, 11 figure
First Colonization of a Spectral Outpost in Random Matrix Theory
We describe the distribution of the first finite number of eigenvalues in a
newly-forming band of the spectrum of the random Hermitean matrix model. The
method is rigorously based on the Riemann-Hilbert analysis of the corresponding
orthogonal polynomials. We provide an analysis with an error term of order
N^(-2 h) where 1/h = 2 nu+2 is the exponent of non-regularity of the effective
potential, thus improving even in the usual case the analysis of the pertinent
literature. The behavior of the first finite number of zeroes (eigenvalues)
appearing in the new band is analyzed and connected with the location of the
zeroes of certain Freud polynomials. In general all these newborn zeroes
approach the point of nonregularity at the rate N^(-h) whereas one (a stray
zero) lags behind at a slower rate of approach. The kernels for the correlator
functions in the scaling coordinate near the emerging band are provided
together with the subleading term: in particular the transition between K and
K+1 eigenvalues is analyzed in detail.Comment: 32 pages, 8 figures (typo corrected in Formula 4.13); some reference
added and minor correction
Non-intersecting squared Bessel paths: critical time and double scaling limit
We consider the double scaling limit for a model of non-intersecting
squared Bessel processes in the confluent case: all paths start at time
at the same positive value , remain positive, and are conditioned to end
at time at . After appropriate rescaling, the paths fill a region in
the --plane as that intersects the hard edge at at a
critical time . In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as of the correlation kernel at critical time and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.Comment: 53 pages, 15 figure
Non-intersecting squared Bessel paths at a hard-edge tacnode
The squared Bessel process is a 1-dimensional diffusion process related to
the squared norm of a higher dimensional Brownian motion. We study a model of
non-intersecting squared Bessel paths, with all paths starting at the same
point at time and ending at the same point at time . Our
interest lies in the critical regime , for which the paths are tangent
to the hard edge at the origin at a critical time . The critical
behavior of the paths for is studied in a scaling limit with time
and temperature . This leads to a critical
correlation kernel that is defined via a new Riemann-Hilbert problem of size
. The Riemann-Hilbert problem gives rise to a new Lax pair
representation for the Hastings-McLeod solution to the inhomogeneous Painlev\'e
II equation where with
the parameter of the squared Bessel process. These results extend
our recent work with Kuijlaars and Zhang \cite{DKZ} for the homogeneous case
.Comment: 54 pages, 13 figures. Corrected error in Theorem 2.
On Eigenvalues of the sum of two random projections
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N
are two N -by-N random orthogonal projections. We relate the joint eigenvalue
distribution of this matrix to the Jacobi matrix ensemble and establish the
universal behavior of eigenvalues for large N. The limiting local behavior of
eigenvalues is governed by the sine kernel in the bulk and by either the Bessel
or the Airy kernel at the edge depending on parameters. We also study an
exceptional case when the local behavior of eigenvalues of P_N + Q_N is not
universal in the usual sense.Comment: 14 page
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