  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 L{\'e}vy
process. This limiting process is shown not to be a free multiplicative
L{\'e}vy 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.
