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 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 $\mathfrak{S}(N)$-Yang-Mills measure associated with the random walk by transposition on $\mathfrak{S}(N)$. This allows us to restrict our study to random walks on $\mathfrak{S}(N)$: we prove theorems about asymptotics of random walks on $\mathfrak{S}(N)$ 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 $(\lambda_N \subset \mathfrak{S}(N))_{N \in \mathbb{N}}$, we define a notion of convergence for $(\lambda_N)_{N \in \mathbb{N}}$ which implies the convergence in non-commutative distribution and in $\mathcal{P}$-expectation of the $\lambda_N$-random walk to a $\mathcal{P}$-free multiplicative Lévy process. This limiting process is shown not to be a free multiplicative Lévy process and we compute its log-cumulant functional. We give also a criterion on $(\lambda_N)_{N \in \mathbb{N}}$ in order to know if the limit is random or not.
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Submitted on : Friday, October 28, 2016 - 5:44:28 PM
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  • HAL Id : hal-01211302, version 2
  • ARXIV : 1510.01046

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Franck Gabriel. A combinatorial theory of random matrices III: random walks on $\mathfrak{S}(N)$, ramified coverings and the $\mathfrak{S}(\infty)$ Yang-Mills measure. 2016. ⟨hal-01211302v2⟩

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