10 research outputs found

    Moments of random matrices and hypergeometric orthogonal polynomials

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    We establish a new connection between moments of n×nn \times n random matrices XnX_n and hypergeometric orthogonal polynomials. Specifically, we consider moments \mathbb{E}\Tr X_n^{-s} as a function of the complex variable s∈Cs \in \mathbb{C}, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden \textit{et al.}~[F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→∞n\to\infty asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

    Lagrangian Multiform Structures, Discrete Systems and Quantisation

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    Lagrangian multiforms are an important recent development in the study of integrable variational problems. In this thesis, we develop two simple examples of the discrete Lagrangian one-form and two-form structures. These linear models still display all the features of the discrete Lagrangian multiform; in particular, the property of Lagrangian closure. That is, the sum of Lagrangians around a closed loop or surface, on solutions, is zero. We study the behaviour of these Lagrangian multiform structures under path integral quantisation and uncover a quantum analogue to the Lagrangian closure property. For the one-form, the quantum mechanical propagator in multiple times is found to be independent of the time-path, depending only on the endpoints. Similarly, for the two-form we define a propagator over a surface in discrete space-time and show that this is independent of the surface geometry, depending only on the boundary. It is not yet clear how to extend these quantised Lagrangian multiforms to non-linear or continuous time models, but by examining two such examples, the generalised McMillan maps and the Degasperis-Ruijsenaars model, we are able to make some steps towards that goal. For the generalised McMillan maps we find a novel formulation of the r-matrix for the dual Lax pair as a normally ordered fraction in elementary shift matrices, which offers a new perspective on the structure. The dual Lax pair may ultimately lead to commuting flows and a one-form structure. We establish the relation between the Degasperis-Ruijsenaars model and the integrable Ruijsenaars-Schneider model, leading to a Lax pair and two particle Lagrangian, as well as finding the quantum mechanical propagator. The link between these results is still needed. A quantum theory of Lagrangian multiforms offers a new paradigm for path integral quantisation of integrable systems; this thesis offers some first steps towards this theory

    Nouvelles perspectives sur les algèbres de type Askey–Wilson

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    Cette thèse se divise en trois parties qui peuvent être toutes regroupées autour d'une même bannière : l'étude de structures algébriques reliées aux algèbres de type Askey–Wilson. Alors que dans la première partie on s'efforce d'obtenir des interprétations duales (au sens de Howe) de ces algèbres, dans les autres parties on étudie des généralisations de ces algèbres. Des dégénérations de l'algèbre de Sklyanin, générées par des blocs plus fondamentaux que ceux générant les algèbres de type Askey–Wilson, sont étudiées dans la deuxième partie et des généralisations de plus haut rang des algèbres de type Askey–Wilson sont étudiées dans la troisième partie. Dans la première partie, en invoquant la dualité de Howe, deux interprétations duales sont obtenues pour les algèbres de Racah, Bannai–Ito, Askey–Wilson, Higgs, Hahn, qq-Hahn et dual −1-1 Hahn. La façon dont la dualité de Howe opère est rendue explicite par l'examen de processus de réduction dimensionnelle. Un modèle superintégrable 2D de mécanique quantique superconforme dont l'algèbre de symétrie est celle de type dual −1-1 Hahn est également introduit et solutionné. Dans la deuxième partie, des algèbres générées par des opérateurs de contiguïté et d'échelle encodant des propriétés de familles de polynômes sont étudiées. Ces opérateurs appartiennent à la classe des opérateurs de Sklyanin–Heun, qui peuvent être définis sur plusieurs grilles diverses. On découvre qu'ils génèrent des dégénérations de l'algèbre de Sklyanin. On démontre que les représentations irréductibles de dimension finie de ces algèbres ont pour base des familles de para-polynômes. Les grilles linéaires, quadratiques, exponentielles et d'Askey–Wilson sont étudiées et mènent respectivement aux polynômes orthogonaux des familles de para-Krawtchouk, para-Racah, qq-para-Krawtchouk et qq-para-Racah. Enfin, la façon dont les polynômes de para-Krawtchouk et d'autres familles de polynômes orthogonaux sont reliées aux représentations tridiagonales du plan de Jordan déformé est présentée. Dans la dernière partie, on explore des généralisations à plus haut rang pour les algèbres de Racah et Askey–Wilson. Pour ce faire, on étudie les réalisations de ces algèbres en termes de Casimirs intermédiaires. Le rôle de la matrice RR tressée est élucidé : celle-ci permet de relier divers Casimirs intermédiaires entre eux par conjugaison. Un isomorphisme entre l'algèbre de skein du crochet de Kauffman de la sphère à 4 trous et l'algèbre engendrée par les Casimir intermédiaires dans Uq(sl2)⊗3U_q(\mathfrak{sl}_2)^{\otimes 3} est présenté et permet d'interpréter de façon diagrammatique la conjugaison par la matrice RR tressée mentionnée ci-haut. Finalement, une présentation du centralisateur Zn(sl2)Z_n(\mathfrak{sl}_2) de U(sl2)U(\mathfrak{sl}_2) dans U(sl2)⊗nU(\mathfrak{sl}_2)^{\otimes n} par générateurs et relations est obtenue et on montre que ce centralisateur est isomorphe à un quotient (obtenu explicitement) de l'algèbre de Racah de plus haut rang R(n)R(n).This thesis is divided in three parts which all orbit around the same theme: the study of algebraic structures related to the algebras of Askey–Wilson type. In the first part we obtain two interpretations that are dual in the sense of Howe for the algebras of Askey–Wilson type. Meanwhile, the other two parts are concerned with generalizations of these algebras. In the second part, we study degenerations of the Sklyanin algebra, which are built out of generators that are more fundamental than those of the Askey–Wilson algebra. In the last part, generalizations of the Askey–Wilson type algebras to higher rank are studied. In the first part, dual interpretations are obtained for the Racah, Bannai–Ito, Askey–Wilson, Higgs, Hahn, qq-Higgs and dual −1-1 Hahn algebras by invoking Howe duality. The way that this Howe duality operates is made explicit through the examination of a dimensional reduction procedure. A 2D superintegrable superconformal quantum mechanics model, whose symmetry algebra is the one of dual −1-1 Hahn type, is also introduced and solved. In the second part, we study algebras that are generated by contiguity and ladder operators that encode properties of families of orthogonal polynomials. We show that these operators belong to the Sklyanin–Heun class of operators, which can be defined for various grids. We also show how their algebraic relations correspond to those of degenerations of the Sklyanin algebra. Then, we show how various families of para-polynomials support finite-dimensional irreducible representations of these degenerate algebras. From the linear, quadratic, exponential and Askey–Wilson grids, we are respectively led to the para-Krawtchouk, para-Racah, qq-para-Krawtchouk and qq-para-Racah polynomials. Later, we connect the para-Krawtchouk polynomials (and other families of orthogonal polynomials) to tridiagonal representations of the deformed Jordan plane. In the final part, we explore higher rank generalizations of the Racah and Askey–Wilson algebras. To that end, their realizations in terms of intermediate Casimir elements are studied. The role of the braided RR-matrix is understood as follows: it connects various intermediate Casimir elements through conjugation. We obtain an isomorphism between the Kauffman bracket skein algebra of the four-punctured sphere and the algebra generated by the intermediate Casimir elements in Uq(sl2)⊗3U_q(\mathfrak{sl}_2)^{\otimes3}. This leads to a diagrammatic interpretation of the conjugation by the braided RR-matrix mentioned in the above. Lastly, a presentation of the centralizer Zn(sl2)Z_n(\mathfrak{sl}_2) of U(sl2)U(\mathfrak{sl}_2) in U(sl2)⊗nU(\mathfrak{sl}_2)^{\otimes n} by generators and relations is obtained and we show that this centralizer is isomorphic to a quotient (which we provide explicitly) of the higher rank Racah algebra R(n)R(n)

    Applications des structures algébriques associées aux systèmes intégrables

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    Cette thèse en trois parties regroupe des travaux de recherches sous la thématiques des symétries sous-jacentes aux systèmes intégrables et des structures algébriques qui les encodent. Une première partie illustre comment les fonctions spéciales que sont les polynômes orthogonaux apparaissent dans la théorie de la représentation des diverses structures algébriques associées à des symétries. La seconde partie se concentre sur une généralisation algébrique de l'opérateur de Heun classique menant à de nouvelles structures algébriques qui trouvent des applications en traitement de signal et dans l'étude des systèmes intégrables. La dernière partie concerne l'élaboration d'un cadre théorique dans le langage de la théorie de l'information algorithmique permettant de poser une définition mathématique de la notion d'émergence.This thesis in three parts groups research work under the theme of the symmetries underlying integrable systems and the algebraic structures that encodes them. A first part illustrates how orthogonal polynomials, a type of special function, appear in the representation theory of various algebraic structures associated to symmetries. The second part focuses on an algebraic generalization of the classical Heun operator that leads to new algebraic structures with applications in signal processing and in the study of integrable systems. The last part concerns the formulation of a framework in the language of algorithmic information theory the enables a mathematical definition for the notion of emergence

    Trans-Series Asymptotics of Solutions to the Degenerate Painlev\'{e} III Equation: A Case Study

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    A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linear auxiliary problem via the isomonodromy deformation approach: trans-series asymptotics for the associated Hamiltonian and principal auxiliary functions and the solution of one of the sigma-forms of the DP3E are also obtained. The actions of Lie-point symmetries for the DP3E are derived.Comment: 102 page

    Acta Scientiarum Mathematicarum : Tomus XXIV. Fasc. 1-2.

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