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Monodromies of ramified random coverings and geometry on partitions

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Random monodromies of natural models of random ramified coverings on the disk define a S(N)- valued planar Markovian holonomy fields. Generalizing a result of T. Levy about the asymptotic, as N goes to infinity, of the Yang-Mills measure which is a special U(N)-valued planar Markovian holonomy field, we prove that the random monodromies of the considered random ramified coverings converge in non-commutative distribution. In order to do so, we will present a proof of the convergence of the eigenvalues distribution of general random walks on the symmetric group which is based on a generalization of ideas from free probability. These new tools were created in order to study random matrices invariant by the symmetric group and are based on a geometric study of partitions.

This talk is part of the Probability series.

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