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37
Nonintersecting paths, random tilings and random matrices
 Probab. Theory Related Fields
, 2002
"... Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the s ..."
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Cited by 78 (8 self)
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Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from nonintersecting Brownian motions. The derivations of the measures are based on the KarlinMcGregor or LindströmGesselViennot method. We use the measures to show some asymptotic results for the models. 1.
Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results
 INT. MATH. RES. NOT
, 2003
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Strong asymptotics for Jacobi polynomials with varying nonstandard parameters
 J. d’Analyse Math
"... Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials P (αn,βn) n is studied, assuming that αn lim n→∞ = A, lim βn n→∞ n n with A and B satisfying A> −1, B> −1, A + B < −1. The asymptotic analysis is based on the nonHermitian orthogonality of these polynomials, and ..."
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Cited by 10 (4 self)
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Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials P (αn,βn) n is studied, assuming that αn lim n→∞ = A, lim βn n→∞ n n with A and B satisfying A> −1, B> −1, A + B < −1. The asymptotic analysis is based on the nonHermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix RiemannHilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case the zeros distribute on the set of critical trajectories Γ of a certain quadratic
Root asymptotics of spectral polynomials for the Lamé operator
 Commun. Math. Phys
"... Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic 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 ..."
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Cited by 9 (4 self)
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Abstract. The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of doubleperiodic 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 Schrödinger 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 integervalued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.
Complex Jacobi matrices
, 1999
"... Complex Jacobi matrices play an important role in the study of asymptotics and zero distribution of Formal Orthogonal Polynomials (FOPs). The latter are essential tools in several fields of Numerical Analysis, for instance in the context of iterative methods for solving large systems of linear eq ..."
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Cited by 7 (4 self)
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Complex Jacobi matrices play an important role in the study of asymptotics and zero distribution of Formal Orthogonal Polynomials (FOPs). The latter are essential tools in several fields of Numerical Analysis, for instance in the context of iterative methods for solving large systems of linear equations, or in the study of Pad'e approximation and Jacobi continued fractions. In this paper we present some known and some new results on FOPs in terms of spectral properties of the underlying (infinite) Jacobi matrix, with a special emphasis to unbounded recurrence coefficients. Here we recover several classical results for real Jacobi matrices. The inverse problem of characterizing properties of the Jacobi operator in terms of FOPs and other solutions of a given three term recurrence is also investigated. This enables us to give results on the approximation of the resolvent by inverses of finite sections, with applications to the convergence of Pad'e approximants. Key words: Dif...
Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform
 J. Approx. Theory
"... The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1asy→∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical p ..."
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Cited by 7 (0 self)
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The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1asy→∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics. Key words and phrases: real Borel probability measure, convergence in distribution, Stieltjes transform, Lévy continuity theorem, AkhiezerKrein theorem, weak convergence of probability measures. Lévy’s classical continuity theorem says that if the pointwise limit of the characteristic functions of a sequence of real Borel probability measures (Pn) exists, then the limit function ϕ is itself the characteristic function for a probability measure P if and only if ϕ is continuous at zero, in which case Pn → P in distribution. The purpose of this note is to prove a direct analog of Lévy’s theorem for Stieltjes transforms, complementing those for other representing functions in [HS] and [HK], and to give several examples of applications. Throughout this note, R and C denote the real and complex numbers, respectively; p.m. and s.p.m. denote Borel probability measures, and subprobability (mass ≤ 1) measures, respectively, on R; and s.p.m.’s (µn) converge vaguely to a s.p.m. µ [C,p.80],ifthere exists a dense subset D of R such that for all a, b ∈ D, a<b,µn((a, b]) → µ((a, b]). (Thus if (µn), µ are p.m.’s, vague convergence is equivalent to convergence in distribution.)
Analysis of nonlinear recurrence relations for the recurrence coefficients of generalized Charlier polynomials
"... The recurrence coe#cients of generalized Charlier polynomials satisfy a system of nonlinear recurrence relations. We simplify the recurrence relations, show that they are related to certain discrete Painleve equations, and analyze the asymptotic behaviour. ..."
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Cited by 6 (4 self)
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The recurrence coe#cients of generalized Charlier polynomials satisfy a system of nonlinear recurrence relations. We simplify the recurrence relations, show that they are related to certain discrete Painleve equations, and analyze the asymptotic behaviour.
ASYMPTOTIC ZERO DISTRIBUTION FOR A CLASS OF MULTIPLE ORTHOGONAL POLYNOMIALS
"... Abstract. We establish the asymptotic zero distribution for polynomials generated by a fourterm recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples o ..."
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Cited by 5 (0 self)
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Abstract. We establish the asymptotic zero distribution for polynomials generated by a fourterm recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular JacobiPiñeiro, Laguerre I and the example associated with Macdonald functions. We also discuss an application to Toeplitz matrices. 1.
RiemannHilbert analysis for Jacobi polynomials orthogonal on a single contour
 J. Approx. Theory
, 2005
"... Classical Jacobi polynomials P (α,β) n, with α, β> −1, have a number of wellknown properties, in particular the location of their zeros in the open interval (−1, 1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong ..."
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Cited by 4 (1 self)
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Classical Jacobi polynomials P (α,β) n, with α, β> −1, have a number of wellknown properties, in particular the location of their zeros in the open interval (−1, 1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters αn, βn depend on n in such a way that αn lim n→ ∞ n = A, lim βn = B, n→ ∞ n with A, B ∈ R. We restrict our attention to the case where the limits A, B are not both