Holonomy fields and random matrices: invariance by braids and permutations.

Abstract : This thesis focuses on planar Yang-Mills measures and planar Markovian holonomy fields. We consider two different questions : the study of planar Markovian holonomy fields with fixed structure group and the asymptotic study of the planar Yang-Mills measures when the dimension of the structure group grows. In the chapter called ``Champs markoviens d'holonomies", we define the notion of planar Markovian holonomy fields which generalizes the concept of planar Yang-Mills measures. We construct, characterize and classify the planar Markovian holonomy fields by introducing a new symmetry : the invariance under the action of braids. In particular, this shows that there is a bijection between planar Markovian holonomy fields and some equivalent classes of Lévy processes. Finally, we use these results in order to characterize Markovian holonomy fields on spherical surfaces. The Markovian holonomy fields with $\mathfrak{S}(N)$ structure group can be constructed using random ramified coverings. We prove in the chapter ``Revêtements ramifiés'' that the monodromies of these models of random ramified coverings converge as the number of sheets of the covering goes to infinity. To prove this, we develop general tools in order to study the limits of families of random matrices invariant by the symmetric group. These tools can be found in the chapter ``Partitions et géométrie'' and in the chapter ``Matrices aléatoires invariantes par le groupe symétrique''. This allows us to generalize ideas, developped by Thierry Lévy in order to study the planar Yang-Mills measure with $U(N)$ structure group, to the setting where the structure group is $\mathfrak{S}(N)$. The chapters ``Partitions et géométrie'' and ``Matrices aléatoires invariantes par le groupe symétrique'', in which we study random matrices invariant by conjugation by some subgroups of the unitary group, can be read independently from the other chapters.
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Franck Gabriel. Holonomy fields and random matrices: invariance by braids and permutations.. Mathematics [math]. Université Paris VI, 2016. English. ⟨tel-01495593⟩

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