Iterated Foldings of Discrete Spaces and Their Limits: Candidates for the Role of Brownian Map in Higher Dimensions

Abstract

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 \(D\hbox {th}\)-random feuilletages (denoted by \(\mathbf{r}[{D}]\)), indexed by a parameter \(D\ge 0\) 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 \((\mathbf{t}^{(j)}, j\ge 1)\). The \(D\hbox {th}\)-random feuilletage \(\mathbf{r}[{D}]\) is built using \((\mathbf{t}^{(1)},\ldots ,\mathbf{t}^{(D)})\): \(\mathbf{r}[{0}]\) is a deterministic circle, \(\mathbf{r}[{1}]\) is Aldous’ continuum random tree, \(\mathbf{r}[{2}]\) is the Brownian map, and somehow, \(\mathbf{r}[{D}]\) is obtained by quotienting \(\mathbf{t}^{(D)}\) by \(\mathbf{r}[{D-1}]\). A discrete counterpart to \(\mathbf{r}[{D}]\) is introduced and called the \(D\)th random discrete feuilletage with \(n+D\) nodes (\(\mathbf{R}_{n}[D]\)). The proof of the convergence of \(\mathbf{R}_{n}[D]\) to \(\mathbf{r}[{D}]\) 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 \(\mathbf{R}_{n}[D]\) is \(n^{1/2^{D}}\). Some elements allowing to conjecture that the Hausdorff dimension of \(\mathbf{r}[{D}]\) is \(2^D\) are given.