published_or_final_version / Physics / Doctoral / Doctor of Philosophy

M.Sc. (Theoretical Physics) / Please refer to full text to view abstract

Submitted by VICTOR CESAR COSTA ALVES null (victorc@ift.unesp.br) on 2017-03-24T17:06:35Z No. of bitstreams: 1 document.pdf: 511663 bytes, checksum: 1bf722030b47e34e0031fc461efd9f67 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-24T20:35:37Z (GMT) No. of bitstreams: 1 alves_vcc_me_ift.pdf: 511663 bytes, checksum: 1bf722030b47e34e0031fc461efd9f67 (MD5) / Made available in DSpace on 2017-03-24T20:35:37Z (GMT). No. of bitstreams: 1 alves_vcc_me_ift.pdf: 511663 bytes, checksum: 1bf722030b47e34e0031fc461efd9f67 (MD5) Previous issue date: 2017-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O presente trabalho trata de um abordagem de aplicações em física dos métodos matemáticos de integrabilidade de Painlevé, por outro lado também aborda o formalismo de hierarquias integráveis e o modelo de 2M-bosons onde são usados métodos de equações diferenciais bem como um método para soluções usando aproximantes de Padé. / The current work aims at applications of mathematical methods of Painlevé integrability in physics, on the other side it also approaches the integrable hierarchies formalism and the 2M-bose model where differential equations methods are used as well as a method for solutions using Padé approximants.

Cette thèse a pour but de résoudre certains problèmes de connexion et de décrire diverses propriétés asymptotiques des fonctions de Painlevé V et I. Dans le cas de l’équation de Painlevé V, nous approchons ces problèmes en développant une nouvelle approche basée sur la théorie conforme des champs bidimensionelle. Nous proposons de calculer les blocs conformes irréguliers de première et seconde espèce par confluence des blocs conformes réguliers de Virasoro. Une conséquence de cette construction est la solution du problème de connexion de l’équation de Painlevé V entre 0 et +i∞. Les formules pour les normalisations relatives (constantes de connexion) de la fonction tau de Painlevé V entre 0, +∞, et +i∞ sont également proposées. Enfin, le développement asymptotique complet de la fonction tau à courte distance pour des données de monodromie génériques est prouvé. Ce résultat est obtenu en construisant une représentation de la fonction tau en termes d’un déterminant de Fredholm. Dans le cas de l’équation de Painlevé I, nous présentons les constantes de connexion relatant les asymptotiques de la fonction tau sur les cinq raies canoniques à l’infini. Ce résultat est obtenu en construisant une extension de la forme différentielle de Jimbo-Miwa-Ueno à l’espace des données de monodromie. Ces constantes de connexion sont exprimées en termes de dilogarithmes de coordonnées de type cluster dans l’espace des données de Stokes. / The aim of this thesis is to solve several connection problems and describe asymptotic properties of Painlevé V and I functions. In the case of Painlevé V equation, we approach these problems by developing a new toolbox based on two dimensional conformal field theory. We propose to compute irregular conformal blocks of the first and second kind by confluence of regular Virasoro conformal blocks. One consequence of this construction is the solution of the connection problem for Painlevé V equation between 0 and +i∞. Formulas for the relative normalizations (connection constants) of Painlevé V tau function between 0, +∞, and +i∞ are also proposed. Finally, the full asymptotic expansion of the tau function at short distances for generic monodromy data is proved. This result is obtained by constructing a Fredholm determinant representation for the tau function. In the case of Painlevé I equation, we present connection constants relating asymptotics of the tau function on the five canonical rays at infinity. This result is obtained by extending the definition of the Jimbo-Miwa-Ueno differential to the space of monodromy data. These connection constants are expressed in terms of dilogarithms of cluster type coordinates on the space of Stokes data.

Indiana University-Purdue University Indianapolis (IUPUI) / We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence. For Painlev´e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians. We use these identities to solve the connection problem for generic solution of Painlev´e-III(D8) equation, and homogeneous Painlev´e-II equation. We formulate conjectures on Hamiltonian and symplectic structure of general isomonodromic deformations we obtained during our studies and check them for Painlev´e equations.

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis. Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE. We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator. In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter. Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here. This thesis is based in part on work to be published jointly with Bálint Virág.

On considère des germes de champs de vecteurs holomorphes singuliers trimimensionnels, appelés noeud-cols doublement résonants. Ces champs de vecteurs correspondent à des systèmes différentiels bidimensionnels à singularité irrégulière, et dont la partie linéaire possède deux valeurs propres non-nulles opposées. Ce type de singularité apparait par exemple à l'infini dans les équations de Painlevé PI,...,PV après compactification à poids de l'espace, pour des valeurs génériques des paramètres. Depuis Boutroux, l'étude de ces singularités a générè de nombreux travaux de recherche. Récemment, plusieurs auteurs ont fournis des informations nouvelles, en étudiant notamment les phénomènes de Stokes non-linéaires et quasi-linéaires associés, en donnant des formules de connexion. Les coefficients de Stokes quasi-linéaires sont invariants sous l'action de changement de coordonnées analytiques locaux, mais ne forment pas un système complet d'invariants analytiques. L'objectif de ce travail de thèse est de fournir une classification analytique générale et complète des noeud-cols doublement résonants. L'idée pour cela est d'adapter les travaux de Martinet et Ramis, généralisés ensuite par Stolovitch. Dans une première partie on fournit une classification formelle, i.e. sous l'action de changements de coordonnées formels, en exhibant des formes normales formelles. Dans un second temps, on étudiera l'existence de normalisations sectorielles (analytiques sur des secteurs), généralisant ainsi un théorème de Hukuhara-Kimura-Matuda. Enfin, on étudiera les recollements entre ces applications normalisantes dans les domaines d'intersections: c'est ce que l'on appellera les difféomorphismes de Stokes. Il s'agira là d'étudier des isotropies sectorielles de la forme normale. On verra que la donnée d'une forme normale formelle et d'un couple de difféomorphismes de Stokes fournira un système complet d'invariants analytiques. Enfin, dans une quatrième et dernière partie, nous calculerons certains de ces invariants pour la singularité irrégulière à l'infini de la première équation de Painlevé. / We consider germs of analytic singular vector fields in dimension three, called doubly-resonant saddle-nodes. These vector fields correspond to irregular two-dimensional systems with a pair of two opposite non-zero eigenvalues. This king of singularity appears for instance at infinity in Painlevé equations PI,...,PV, after a weighted compactifcation, for generic values of the parameters. Since Boutroux, the study of these singularities has generated many researches. Recently, several authors provided new informations, by studying for instance the associated non-linear and quasi-lineair Stokes phenomenas and by giving connection formulas. Quasi-linéaire Stokes coefficients are invariant under local analytic change of coordinates, but do not form a complete set of invariants for analytic classification. The goal of this work is to provide a complete analytic classification of doubly-resonant saddle-nodes. The idea for this is to adapt the works of Martinet and Ramis, generalized then by Stolovitch. In the first part, we give a formal classification, based on the existence on unique formal normal forms. In the second part, we prove the existence of sectorial nomalizing maps (analytic over sectors), generalizing a theorem by Hukuhara-Kimura-Matuda. In the third part, we study the Stokes diffeomorphisms, and more generaly the sectorials isotropies of the normal form. We obtain a complet set of analytic invariants. Finally, in the fourth part, we compute some of these invariants in the case of the first Painlevé equation.