Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes
Résumé
In this article, we study random matrices in a framework based on the geometric study of partitions and some dualities as the Schur-Weyl's duality. This gives a unified and simple framework in order to understand families of random matrices which are invariant by conjugation in law by any group whose associated tensor category is spanned by partitions. This includes the unitary groups, the unitary reflection groups, the orthogonal groups, the bistochastic groups, the hyperoctahedral groups and the symmetric groups. For each choice of symmetry one can associate a subset of partitions A which allows us to define a notion of A-free cumulants. Besides, we introduce some observables on random matrices, namely the P-moments, which generalize the normalized moments. One of the various by-products we get is that for any family of random matrices which is invariant by the unitary group, if it converges in non-commutative distribution then the P-moments of this family converge in expectation. This implies a simple formula which allows us to compute the asymptotic of any product of the entries of a family of random matrices which is invariant in law by conjugation by the unitary group and which converges in non-commutative distribution. We prove similar results when the family is invariant in law by conjugation by the orthogonal group. This setting leads to a unified way in order to define and study new notions of asymptotic freeness associated to each symmetry. As a by-product, we prove that independence and invariance in law by conjugation by the bistochastic group implies asymptotic Voiculescu's freeness. We show that there exist two formulations for each notion of asymptotic freeness: one uses some modified moments, and the other uses cumulants. In this setting, a non-commutative central limit theorem is proved and the notion of asymptotic factorization is also studied. We also show how to inject the theory of classical probabilities in this new framework: as a consequence, the classical cumulants can be seen as a special case of the new free cumulants we defined. At last, we give general theorems about convergence of matrix-valued additive and multiplicative Levy processes which are invariant in law by conjugation by the symmetric group. Using these results, we give a unified point of view on the study of matricial Levy processes on some Lie algebras and some Lie groups.
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