Strong Asymptotics for Multiple Laguerre Polynomials

Лысов, Владимир Генрихович; Wielonsky, Franck

doi:10.1007/s00365-006-0648-1

Cited by 28 publications

(51 citation statements)

References 24 publications

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“…Such a construction was given before for a 2 × 2 RH problem in [40] and in a 3 × 3 setting in [38,43]. Our construction will be similar to the one in [38].…”

Section: Parametrix Near the Branch Point 0 (Hard Edge)mentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T . After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

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“…Such a construction was given before for a 2 × 2 RH problem in [40] and in a 3 × 3 setting in [38,43]. Our construction will be similar to the one in [38].…”

Section: Parametrix Near the Branch Point 0 (Hard Edge)mentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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“…To motivate further the importance of an asymptotic description for the zeros, let us mention that they play an important role in the strong asymptotic for MOPs. For example, the work [41] of Lysov and Wielonsky deals with the strong asymptotics of multiple Laguerre polynomials in the case where r = 2. A key ingredient in their analysis is the a priori knowledge of an algebraic equation for the Cauchy-Stieltjes transform of the limiting zero distribution (that they denote ψ(z), up to a trivial rescaling), see equation [41, (1.4)].…”

Section: Application To Multiple Orthogonal Polynomialsmentioning

confidence: 99%

Average characteristic polynomials of determinantal point processes

Hardy¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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We investigate the average characteristic polynomial E N i=1 (z −x i ) where the x i 's are real random variables drawn from a Biorthogonal Ensemble, i.e. a determinantal point process associated with a bounded finite-rank projection operator. For a subclass of Biorthogonal Ensembles, which contains Orthogonal Polynomial Ensembles and (mixed-type) Multiple Orthogonal Polynomial Ensembles, we provide a sufficient condition for its limiting zero distribution to match with the limiting distribution of the random variables, almost surely, as N goes to infinity. Moreover, such a condition turns out to be sufficient to strengthen the mean convergence to the almost sure one for the moments of the empirical measure associated to the determinantal point process, a fact of independent interest. As an application, we obtain from Voiculescu's theorems the limiting zero distribution for multiple Hermite and multiple Laguerre polynomials, expressed in terms of free convolutions of classical distributions with atomic measures, and then derive explicit algebraic equations for their Cauchy-Stieltjes transform. AbstractOn s'intéresse au polynôme caractéristique moyen E N i=1 (z −x i ) associéà des variables aléatoires réelles x 1 , . . . , x N qui forment un Ensemble Biorthogonal, c'est-à-dire un processus ponctuel déterminantal associéà un opérateur de projection borné et de rang fini. Pour une sous-classe d'Ensembles Biorthogonaux, qui contient les Ensembles Polynômes Orthogonaux et les Ensembles Polynômes Orthogonaux Multiples (de type mixte), nous obtenons une condition suffisante pour que, presque sûrement, la distribution limite de ses zéros coincide avec la distribution limite des variables aléatoires, quand N tend vers l'infini. De plus, cette condition s'avèreêtreégalement suffisante pour améliorer la convergence en moyenne en convergence presque sûre pour les moments de la mesure empirique associée au processus ponctuel déterminantal. En application, on obtient avec des théorèmes de Voiculescu une description pour les distributions limites des zéros des polynômes d'Hermite et de Laguerre multiples, en termes de convolutions libres de lois classiques avec des mesures atomiques, ainsi que deséquations algébriques explicites pour leurs transformées de Cauchy-Stieltjes.

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“…Fortunately, this kind of behavior has been analyzed in [41] (for a 2 × 2 matrix valued RH problem), and in [45] (for a 3 × 3 matrix valued RH problem) and we use the construction from these papers. There will be a new feature though in the case −1 < α < 0.…”

Section: P Is Bounded Asmentioning

confidence: 99%

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Kuijlaars

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

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120

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We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t * the smallest paths hit the hard edge and from then on are stuck to it. For t = t * 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 t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

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Strong Asymptotics for Multiple Laguerre Polynomials

Cited by 28 publications

References 24 publications

Non-intersecting squared Bessel paths with one positive starting and ending point

Non-intersecting squared Bessel paths with one positive starting and ending point

Average characteristic polynomials of determinantal point processes

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

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