67 research outputs found

    Matricial approximations of higher dimensional master fields

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    We study matricial approximations of master fields constructed in [6]. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in R, C or K) and letting the dimension of these blocks to tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described

    Symétries de jauge non-commutative et diffusions pseudo-unitaires

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    This thesis is devoted to the study of two quite different questions, which are related by the tools that we used to study them.The first question is that of the definition of lattice gauge theories with a non-commutative structure group. Here, by non-commutative, we do not mean non-Abelian, but instead non-commutative in the general sense of non-commutative geometry.The second question is that of the behaviour of Brownian diffusions on non-compact matrix groups of a specific kind, namely groups of pseudo-orthogonal, pseudo-unitary or pseudo-symplectic matrices.In the first chapter, we investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a Zhang algebra. Zhang algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups.We are interested in non-commutative analogues of random gauge fields, which we describe through the random holonomy that they induce.We propose a general definition of a holonomy field with Zhang gauge symmetry, and construct such a field starting from a quantum Lévy process on a Zhang algebra.As an application, we define higher dimensional generalizations of the so-called master field.In the second chapter, we study matricial approximations of higher dimensional master fields constructed in the previous chapter.These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in the algebras of real, complex or quaternionic numbers) and letting the dimension of these blocks tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks.In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described.In the last two chapters, we use tools introduced (Zhang algebras and coloured Brauer diagrams) in the first two ones to study Brownian motion on pseudo-unitary matrices in high dimensions.We prove convergence in non-commutative distribution of the pseudo-unitary Brownian motions we consider, to free with amalgamation semi-groups under the hypothesis of convergence of the normalized signature of the metric.In the split case, meaning that at least asymptotically the metric has as much negative directions as positive ones, the limiting distribution is that of a free L\'evy process, which is a solution of a free stochastic differential equation.We leave open the question of such a realization of the limiting distribution in the general case.In addition, we provide (intriguing) numerical evidences for the convergence of the spectral distribution of such random matrices and make two conjectures. At the end of the thesis, we prove asymptotic normality for the fluctuations.Cette thÚse est consacrée à l'étude de deux questions trÚs différentes, reliées par les outils que nous utilisons pour les étudier. La premiÚre question est celle de la définition des théories de jauge sur un réseau avec un groupe de structure non commutatif. Ici, non commutatif ne signifie pas non Abelian, mais plutÎt non commutatif au sens général de la géométrie non commutative. La deuxiÚme question est celle du comportement des diffusions Browniennes sur des groupes matriciels non compacts d'un type spécifique, à savoir des groupes de matrices pseudo-orthogonales, pseudo-unitaires ou pseudo-symplectiques.Dans le premier chapitre, nous étudions des théories de jauge quantiques sur un réseau et leur limite continue sur le plan euclidien ayant une algÚbre de Zhang pour groupe de stucture. Les algÚbres de Zhang sont des analogues non commutatifs des groupes et contiennent la classe des groupes duaux de Voiculescu. Nous nous intéressons donc aux analogues non commutatifs des champs de jauges quantiques, que nous décrivons par l'holonomie aléatoire qu'ils induisent. Nous proposons une définition générale d'un champ d'holonomies ayant une symétrie de jauge présentant la structure d'une algÚbre de Zhang, et construisons un tel champ à partir d'un processus quantique de Lévy sur une algÚbre de Zhang.Dans le deuxiÚme chapitre, nous étudions les approximations matricielles des champs maßtres en dimensions supérieures construits dans le chapitre précédent. Ces approximations (en distribution non commutative) sont obtenues en extrayant des blocs d'une diffusion unitaire Brownienne (à coefficients dans les algÚbres de nombres réels, complexes ou quaternioniques) et en laissant la dimension de ces blocs tendre vers l'infini. Nous divisons notre étude en deux parties : dans la premiÚre, nous extrayons des blocs carrés tandis que dans la seconde, nous autorisons des blocs rectangulaires.Dans les deux derniers chapitres, nous utilisons les outils introduits (algÚbres de Zhang et diagrammes de Brauer colorés) dans les deux premiers pour étudier des diffusions sur des groupes de matrices pseudo-unitaires. Nous prouvons la convergence non commutative des mouvements Browniens pseudo-unitaires que nous considérons vers des semi-groupes libres avec amalgamation sous l'hypothÚse de convergence de la signature normalisée de la métrique de l'espace sous-jacent.Dans le cas déployé, c'est-à-dire, qu'au moins asymptotiquement, la métrique a autant de directions négatives que de directions positives, la distribution limite est la distribution d'un processus de Lévy, solution d'une équation différentielle stochastique libre. Nous laissons ouverte la question d'une telle réalisation de la distribution limite dans le cas général.De plus, nous présentons des résultats numériques sur la convergence de la distribution spectrale de ces matrices aléatoires et faisons deux conjectures. Dans le dernier chapitre, nous prouvons la normalité asymptotique des fluctuations

    Combinatorics of cyclic-conditional freeness

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    We develop the relevant combinatorics pertaining to cyclic-conditional freeness in order to introduce the adequate sets of cumulants linearizing the cyclic-conditional additive convolution. On our way, we introduce a new non-commutative independence, cyclic freeness. We explain how cyclic-conditional freeness is "reduced" to cyclic freeness by utilizing a multivariate extension of the inverse Markov-Krein transform. We also consider cyclic-conditional multiplicative convolution and prove limit theorems

    Influence of Hydrocarbons Exposure on Survival, Growth and Condition of Juvenile Flatfish

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    Juveniles of numerous commercial marine flatfish species use coastal and estuarine habitats as nurseries. Hence, they are likely to be exposed to a number of anthropogenic stressors such as accidental and chronic exposure to chemical contaminants. Little is known about their response to such pollutants at the individual level and about the consequences on their population dynamics. Mesocosm experiments were conducted to determine whether short (24 h) but high exposure to petroleum hydrocarbons (1/1000 v: v water: fuel), similar to what happened after an oil spill on coastal areas, af fects survival and biological (growth, body condition and lipid reserve) performances of juvenile common sole, which live on near shore and estuarine nursery grounds. Results demonstrated that this type of exposure significantly reduce survival, growth (size, recent otolith increment and body condition), and especially ener gy storage (triacylglycerol to free sterol ratio) of the juvenile fish on the medium-term (three months after the exposure).These medium-term consequences affect future recruitment of this long- lived species

    Crecimiento y condiciĂłn de juveniles de lenguado (Solea solea L.) como indicadores de calidad de hĂĄbitat en ĂĄreas de crĂ­a costeras y estuĂĄricas del Golfo de Vizcaya con Ă©nfasis en zonas expuestas al vertido del Erika

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    Indicators of growth and condition were used to compare the habitat quality of nurseries of juvenile sole (Solea solea L.) in the Bay of Biscay, based on one survey in 2000. The four biological indicators are poorly correlated with each other, suggesting that no single measure may give an adequate description of fish health and of its habitat’s quality. Growth indicators showed significant differences between northern and southern areas. Juveniles from the two southernmost nurseries, the Gironde estuary and the Pertuis Antioche, displayed significant lower otolith increment widths and mean sizes. These differences were inversely related to water temperature and unrelated to genetic or age differences, and are unlikely to be due to limiting trophic conditions in the nurseries. Hence, they may be considered in terms of differences in habitat quality and potential anthropogenic impacts. Condition indices do not show this north-south pattern but highlight low condition values in the Pertuis Antioche. Short-term and fluctuating biochemical indicators such as RNA/DNA ratios appeared to be unreliable over a long-term study, while morphometric indices seemed to be relevant, complementary indicators as they integrate the whole juvenile life-history of sole in the nurseries. The growth and condition indices of juveniles in September 2000 from nursery grounds exposed to the Erika oil spill in December 1999 were relatively high. These results lead us to suggest that there was no obvious impact of this event on the health of juvenile sole and on the quality of the exposed nursery grounds a few months after the event.Se usaron indicadores de crecimiento y condición de lenguados juveniles (Solea solea L.) para comparar la calidad del hábitat de áreas de cría en el Golfo de Vizcaya basados en un muestreo de 2000. Los cuatro indicadores biológicos mostraron bajas correlaciones entre sí, lo que sugiere que no existe una medida única para describir adecuadamente el estado de los peces y la calidad de su hábitat. Los indicadores de crecimiento mostraron diferencias significativas entre las áreas septentrionales y meridionales. Los juveniles de las áreas situadas más al sur: el estuario del Garona y Pertuis Antioche mostraron incrementos de otolitos y tamaños medios significativamente inferiores. Estas diferencias están inversamente relacionadas con la temperatura del agua, pero no con diferencias genéticas o de edad, y no es probable que se deba a una condición trófica limitante en las áreas de cría. Por tanto, pueden considerarse en términos de calidad del hábitat y el potencial impacto antropogénico. Los índices de condición no muestras este patrón norte-sur sino que destacan los bajos valores de condición en Pertuis Antioche. Los indicadores bioquímicos variables de corto término como las relaciones RNA/DNA no parecieron fiables para estudios de mayor escala, mientras que los índices morfométricos parecen ser relevantes y complementarios ya que integran toda el período juvenil de la vida de los lenguados en sus áreas de cría. Los valores de crecimiento y condición de juveniles en septiembre de 2000 en áreas de cría expuestas al vertido de crudo del Erika en diciembre de 1999 fueron relativamente elevadas. Estos resultados sugieren que no hubo un impacto obvio de este evento sobre la salud de los juveniles de lenguado ni en la calidad de las áreas de cría expuestas transcurridos unos pocos meses después del vertido

    Symétrie de jauge non commutative et diffusions pseudo-unitaires

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    This thesis is devoted to the study of two quite different questions, which are related by the tools that we use to study them. The first question is that of the definition of lattice gauge theories with a non-commutative structure group. Here, by non-commutative, we do not mean non-Abelian, but instead non-commutative in the general sense of non-commutative geometry. The second question is that of the behaviour of Brownian diffusions on non-compact matrix groups of a specific kind, namely groups of pseudo-orthogonal, pseudo-unitary or pseudo-symplectic matrices. In the first chapter, we investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a Zhang algebra. Zhang algebras are non-commutative analogues of groups and contain the class of Voiculescu’s dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random holonomy that they induce. We propose a general definition of a holonomy field with Zhang gauge symmetry, and construct such a field starting from a quantum LĂ©vy process on a Zhang algebra. As an application, we define higher dimensional generalizations of the so-called master field. In the second chapter, we study matricial approximations of higher dimensional master fields constructed in the previous chapter. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in the algebras of real, complex or quaternionic numbers) and letting the dimension of these blocks tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described. In the last two chapters, we use tools introduced (Zhang algebras and coloured Brauer diagrams) in the first two ones to study Brownian motion on pseudo-unitary matrices in high dimensions. We prove convergence in non-commutative distribution of the pseudo-unitary Brownian motions we consider to free with amalgamation semi-groups under the hypothesis of convergence of the normalized signature of the metric. In the split case, meaning that at least asymptotically the metric has as much negative directions as positive ones, the limiting distribution is that of a free LĂ©vy process, which is a solution of a free stochastic differential equation. We leave open the question of such a realization of the limiting distribution in the general case. In addition we provide (intriguing) numerical evidences for the convergence of the spectral distribution of such random matrices and make two conjectures. At the end of the thesis, we prove asymptotic normality for the fluctuations.Cette thĂšse est consacrĂ©e Ă  l’étude de deux questions trĂšs diffĂ©rentes, reliĂ©es par les outils que nous utilisons pour les Ă©tudier. La premiĂšre question est celle de la dĂ©finition des thĂ©ories de jauge sur un rĂ©seau avec un groupe de structure non commutatif. Ici, non commutatif ne signifie pas non Abelian, mais plutĂŽt non commutatif au sens gĂ©nĂ©ral de la gĂ©omĂ©trie non commutative. La deuxiĂšme question est celle du comportement des diffusions Browniennes sur des groupes matriciels non compacts d’un type spĂ©cifique, Ă  savoir des groupes de matrices pseudo-orthogonales, pseudo-unitaires ou pseudo-symplectiques. Dans le premier chapitre, nous Ă©tudions des thĂ©ories de jauge quantiques sur un rĂ©seau et leur limite continue sur le plan euclidien ayant une algĂšbre de Zhang pour groupe de stuc-ture. Les algĂšbres de Zhang sont des analogues non commutatifs des groupes et contiennent la classe des groupes duaux de Voiculescu. Nous nous intĂ©ressons donc aux analogues non commutatifs des champs de jauges quantiques, que nous dĂ©crivons par l’holonomie alĂ©atoire qu’ils induisent. Nous proposons une dĂ©finition gĂ©nĂ©rale d’un champ d’holonomies ayant une symĂ©trie de jauge prĂ©sentant la structure d’une algĂšbre de Zhang, et construisons un tel champ Ă  partir d’un processus quantique de LĂ©vy sur une algĂšbre de Zhang. Dans le deuxiĂšme chapitre, nous Ă©tudions les approximations matricielles des champs maĂźtres en dimensions supĂ©rieures construits dans le chapitre prĂ©cĂ©dent. Ces approximations (en distribution non commutative) sont obtenues en extrayant des blocs d’une diffusion unitaire Brownienne (Ă  coefficients dans les algĂšbres de nombres rĂ©els, complexes ou quaternioniques) et en laissant la dimension de ces blocs tendre vers l’infini. Nous divisons notre Ă©tude en deux parties : dans la premiĂšre, nous extrayons des blocs carrĂ©s tandis que dans la seconde, nous autorisons des blocs rectangulaires. Dans les deux derniers chapitres, nous utilisons les outils introduits (algĂšbres de Zhang et diagrammes de Brauer colorĂ©s) dans les deux premiers pour Ă©tudier des diffusions sur des groupes de matrices pseudo-unitaires. Nous prouvons la convergence non commutative des mouvements Browniens pseudo-unitaires que nous considĂ©rons vers des semi-groupes libres avec amalgamation sous l’hypothĂšse de convergence de la signature normalisĂ©e de la mĂ©trique de l’espace sous-jacent. Dans le cas dĂ©ployĂ©, c’est-Ă -dire, qu’au moins asymptotiquement, la mĂ©trique a autant de directions nĂ©gatives que de directions positives, la distribution limite est la distribution d’un processus de LĂ©vy, solution d’une Ă©quation diffĂ©rentielle stochastique libre. Nous laissons ouverte la question d’une telle rĂ©alisation de la distribution limite dans le cas gĂ©nĂ©ral. De plus, nous prĂ©sentons des rĂ©sultats numĂ©riques sur la convergence de la distribution spectrale de ces matrices alĂ©atoires et faisons deux conjectures. Dans le dernier chapitre, nous prouvons la normalitĂ© asymptotique des fluctuations
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