A combinatorial theory of random matrices III: random walks on $\mathfrak{S}(N)$, ramified coverings and the $\mathfrak{S}(\infty)$ Yang-Mills measure

Abstract

The aim of this article is to study some asymptotics of a natural model of random ramified coverings on the disk of degree N . We prove that the monodromy field, called also the holonomy field, converges in probability to a non-random continuous field as N goes to infinity. In order to do so, we use the fact that the monodromy field of random uniform labelled simple ramified coverings on the disk of degree N has the same law as the S(N )-Yang-Mills measure associated with the random walk by transpositions on S(N ).This allows us to restrict our study to random walks on S(N ): we prove theorems about asymptotics of random walks on S(N ) in a new framework based on the geometric study of partitions and the Schur-Weyl-Jones's dualities. In particular, given a sequence of conjugacy classes (λN ⊂ S(N )) N∈N , we define a notion of convergence for (λN ) N∈N which implies the convergence in non-commutative distribution and in P-distribution of the λN -random walk to a multiplicative P-Lévy process. This limiting process is shown not to be a free multiplicative Lévy process and we compute its log-cumulant transform. We give also a criterion on (λN ) N∈N in order to know if the limit is random or not.

Hidden symmetries and large N factorisation for permutation invariant matrix observables

“…This is the first sense in which hidden symmetries appear in this paper. Permutation invariant random matrix distributions have also been studied from the point of view of mathematical statistics, using partition algebra diagrams [44][45][46].…”

Large permutation invariant random matrices are asymptotically free over the diagonal

“…One can accommodate such models by defining a new framework. This is the perspective of traffic probability [2,10,12,13,14,16,17,18].…”

Traffic distributions and independence II: Universal constructions for traffic spaces

“…Remark that Gabriel [12] proved independently a result similar to Theorem 1.1 about the convergence of permutation invariant observables on random matrices. More generally, up to some conventions the framework developed in [11][12][13] is equivalent to the framework of traffics. Interestingly, it develops aspects that are not yet considered for traffics, such as the central notion of cumulants.…”