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

TL;DR

This paper introduces a sequence of random geometric objects called $D$th-random feuilletages, which generalize the Brownian map to higher dimensions, and studies their properties and convergence behaviors.

Contribution

It constructs higher-dimensional analogues of the Brownian map using iterated Brownian snakes and establishes their convergence from discrete models.

Findings

Convergence of discrete feuilletages to continuous limits after rescaling

Upper bound on the diameter of discrete models as n^{1/2^D}

Conjecture that Hausdorff dimension of ${f r}[D]$ is 2^D

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 $D$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.