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The aim of this article is to study some asymptotics of a natural model ofrandom ramified coverings on the disk of degree $N$. We prove that themonodromy field, called also the holonomy field, converges in probability to anon-random field as $N$ goes to infinity. In order to do so, we use the factthat the monodromy field of random uniform labelled simple ramified coveringson the disk of degree $N$ has the same law as the $\mathfrak{S}(N)$-Yang-Millsmeasure associated with the random walk by transposition on $\mathfrak{S}(N)$.This allows us to restrict our study to random walks on $\mathfrak{S}(N)$: weprove theorems about asymptotics of random walks on $\mathfrak{S}(N)$ in a newframework based on the geometric study of partitions and the Schur-Weyl-Jones'sdualities. In particular, given a sequence of conjugacy classes $(\lambda\_N\subset \mathfrak{S}(N))\_{N \in \mathbb{N}}$, we define a notion ofconvergence for $(\lambda\_N)\_{N \in \mathbb{N}}$ which implies theconvergence in non-commutative distribution and in $\mathcal{P}$-expectation ofthe $\lambda\_N$-random walk to a $\mathcal{P}$-free multiplicative L{\'e}vyprocess. This limiting process is shown not to be a free multiplicativeL{\'e}vy process and we compute its log-cumulant functional. We give also acriterion on $(\lambda\_N)\_{N \in \mathbb{N}}$ in order to know if the limitis random or not.

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