  In this last decade, an important stochastic model emerged: the Brownian map.
It is the limit of various models of random combinatorial maps after rescaling:
it is a random metric space with Hausdorff dimension 4, almost surely
homeomorphic to the 2-sphere, and possesses some deep connections with
  Liouville quantum gravity in 2D. In this paper, we present a sequence of
random objects that we call $D$th-random feuilletages (denoted by ${\bf
r}[D]$), indexed by a parameter $D\geq 0$ and which are candidate to play the
role of the Brownian map in dimension $D$. The construction relies on some
objects that we name iterated Brownian snakes, which are branching analogues of
iterated Brownian motions, and which are moreover limits of iterated discrete
snakes. In the planar $D=2$ case, the family of discrete snakes considered
coincides with some family of (random) labeled trees known to encode planar
quadrangulations.
  Iterating snakes provides a sequence of random trees $({\bf t}^{(j)}, j\geq
1)$. The $D$th-random feuilletage ${\bf r}[D]$ is built using $({\bf
t}^{(1)},\cdots,{\bf t}^{(D)})$: ${\bf r}[0]$ is a deterministic circle, ${\bf
r}[1]$ is Aldous' continuum random tree, ${\bf r}[2]$ is the Brownian map, and
somehow, ${\bf r}[D]$ is obtained by quotienting ${\bf t}^{(D)}$ by ${\bf
r}[D-1]$.
  A discrete counterpart to ${\bf r}[D]$ is introduced and called the $D$th
random discrete feuilletage with $n+D$ nodes (${\bf r}_n[D]$). The proof of the
convergence of ${\bf r}_n[D]$ to ${\bf r}[D]$ after appropriate rescaling in
some functional space is provided (however, the convergence obtained is too
weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter
of ${\bf r}_{n}[D]$ is $n^{1/2^{D}}$. Some elements allowing to conjecture that
the Hausdorff dimension of ${\bf r}[D]$ is $2^D$ are given.
