Numerical estimation of the Hausdorff dimension of D-random feuilletages

TL;DR

The work numerically investigates higher-dimensional random geometries built from iterated folding of trees, focusing on the $D=3$ case to test the conjectured scaling $d_H( extbf{r}[D])=2^D$. Using a finite-size scaling approach on distance histograms, the authors validate their estimator against the exactly known $D=2$ Brownian map case and obtain strong numerical evidence supporting $d_H=8$ for $D=3$, within finite-size uncertainties. The methodology adapts established scaling analyses of random geometries to the $D>2$ setting, providing a practical tool to explore potential universal classes beyond the Brownian map. The results bolster the proposal that $ extbf{R}_n[3]$ converges to a genuine higher-dimensional random geometry with fractal scaling, with potential relevance for quantum gravity discretizations and the study of scaling limits in higher dimensions.

Abstract

We implement numerical techniques to simulate D-random feuilletages, candidates for higher-dimensional random geometries introduced in L. Lionni and J.-F. Marckert, Math. Phys. Anal. Geom. 24 (2021) 39. Using finite-size scaling techniques, our approach allows to give a numerical estimation of the Hausdorff dimension $d_H$ of these feuilletages. The results obtained are compatible with the formal result known for the Brownian map, which corresponds to the D=2 random feuilletage. For the D=3 case, our numerical study finds a good agreement with the conjectured value $d_H=8$.

Figures (12)

• Figure 1: a) Rooted plane tree with $n$ edges, the root vertex is at the bottom. b) Contour function obtained by recording the distance of each vertex from the root vertex. c) Brownian excursion obtained in the scaling limit. An example of the distance identification \ref{['eq:identif_brow_exc']} is shown in horizontal arrows. d) Illustration of the Continuuos Random Tree.
• Figure 2: a) Pointed planar quadrangulation with 6 faces. The root vertex is shown in fuchsia. Each vertex is labeled by its graph distance to the root. b) The spanning tree (in black) is obtained by the CVS bijection with positive labels. c) Labeled plane tree with 5 edges and vertex labels such that between adjacent vertices the labels differ by $-1$, $0$ or $+1$.
• Figure 3: a) Uniform labeled plane tree with $5$ edges $\mathbf{T}_5^1$. b) Label process of $\mathbf{T}_5^1$. c) Conjugation of the label process. d) Height process of the random plane tree with $10$ edges, $\mathbf{T}_{10}^2$.
• Figure 4: a) $(D=1)$-random discrete feuilletage $\mathbf{T}^1$. b) Identification of the vertices of the random plane tree $\mathbf{T}^2$ induced by the corners of $\mathbf{T}_1$. c) Corresponding uniform pointed planar quadrangulation or $(D=2)$-random discrete feuilletage. d) Identification of the vertices of the random plane tree $\mathbf{T}^3$ induced by the corners of $\mathbf{T}^2$. e) Corresponding random planar map with two-marked vertices. f) $(D=3)$-random discrete feuilletage obtained from identifying vertices of the random plane trees $\mathbf{T}^3$ (according to the corners of $\mathbf{T}^2$) and $\mathbf{T}^2$ (according to the corners of $\mathbf{T}_1$).
• Figure 5: Left: Distance histograms $\rho_n(x)$ for $n\in [2^{11}, 2^{21}]$. Right: Example of the fit to $\rho_{n_0}$ with $n_0=2^{21}$ from which the numerical values of $k_n$ are obtained.