Iterated foldings of discrete spaces and their limits: candidates for the role of Brownian map in higher dimensions

Understand

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.