D-Random Feuilletages in Random Geometry

• D-random feuilletages are hierarchical random metric spaces defined via iterated Brownian snakes, generalizing structures from the unit circle to the Brownian map.
• Discrete models using Dyck paths and branching random walks converge to continuum structures with conjectured Hausdorff dimensions d_H = 2^D, supported by Monte Carlo simulations.
• Recursive Gaussian processes and equivalence relations underpin the construction, offering a new framework for exploring high-dimensional random geometry and universality classes.

D-random feuilletages ($\mathbf{r}[D]$) constitute a hierarchy of random metric spaces introduced as candidates for the scaling limits of iterated foldings of discrete structures, generalizing the Brownian map ($D=2$) to arbitrary integer dimension $D \geq 0$. Their construction is rooted in the theory of iterated Brownian snakes—branching, recursively defined Gaussian processes—yielding compact spaces that interpolate between the unit circle ($D=0$), the continuum random tree ($D=1$), the Brownian map ($D=2$), and conjecturally, higher-dimensional continuum analogues for $D \geq 3$. Discrete counterparts $\mathbf{r}_n[D]$ provide explicit combinatorial models whose large-$n$ scaling limits are believed to realize the continuum $\mathbf{r}[D]$ (Lionni et al., 2019). Recent work has implemented large-scale Monte Carlo simulations, measuring the Hausdorff dimension $d_H$ for $D=2,3$, with results consistent with the conjecture $d_H(\mathbf{r}[D]) = 2^D$ (Castro et al., 6 Nov 2025).

1. Construction of D-random Feuilletages

The continuous $D$th-random feuilletage $\mathbf{r}[D]$ is constructed through a system of iterated Brownian snakes. The process begins with $h^{(1)}(\cdot) \equiv e(\cdot)$, a normalized Brownian excursion on $[0,1]$. For each $j=2,\dots,D$, a centered Gaussian process $\mathrm{Bell}^{(j-1)}$ is defined, with conditional covariance determined via

$$\mathrm{Cov}\big(\mathrm{Bell}^{(j-1)}(s),\mathrm{Bell}^{(j-1)}(t)\big) = \widecheck{h^{(j-1)}}(s,t), \quad \widecheck{g}(s,t)=\min_{u\in[s\wedge t,\,s\vee t]}g(u).$$

The process $h^{(j)}$ is recursively defined by

$$h^{(j)}(x) = \mathrm{Bell}^{(j-1)}\left(x + A^{(j-1)}\right) - m^{(j-1)},$$

where $A^{(j-1)}$ denotes the minimum point and $m^{(j-1)}$ the minimum value of $\mathrm{Bell}^{(j-1)}$. Each $h^{(j)}$ encodes a compact rooted $\mathbb{R}$-tree $\mathbf{T}^{(j)}$ via the standard "height" construction, and associated metrics

$$D_{h^{(j)}}(x,y) = h^{(j)}(x) + h^{(j)}(y) - 2\widecheck{h^{(j)}}(x,y).$$

Points in $[0,1]$ are identified recursively: for each $m=1,\dots,D$, equivalence relations $\sim_{[m]}$ are generated by $x\sim_{[m]}y$ when $D_{h^{(m)}}(x-A^{(m)},y-A^{(m)})=0$, and $\sim_D$ is the transitive closure of these identifications. The continuous $D$th-random feuilletage is then defined as the measured quotient

$\mathbf{r}[D]=[0,1]/\sim_D.$

Natural metric structures such as the "multi-tree infimum" and "last-tree only" distances can be placed on $\mathbf{r}[D]$, though the Gromov–Hausdorff continuity of these metrics remains unverified (Lionni et al., 2019).

2. Discrete Models and Iterated Foldings

The discrete $D$th-random feuilletage, denoted $\mathbf{r}_n[D]$, is constructed via a sequence of discrete snakes: $(\mathbf{C}_n^{(j)},\mathbf{L}_n^{(j)})$ for $j=1,\dots,D$, where $\mathbf{C}_n^{(j)}$ is a Dyck path (contour of a plane tree $\mathbf{T}_n^{(j)}$ of size $2^{j-1}n$) and $\mathbf{L}_n^{(j)}$ is a branching random walk labeling. For each $j\ge2$, corners of $\mathbf{T}_n^{(j)}$ associated to the same parent-tree corner in $\mathbf{T}_n^{(j-1)}$ are identified, yielding a nested sequence of non-crossing partitions, and ultimately a connected graph with $n+D$ vertices composed from edges of $\mathbf{T}_n^{(D)}$.

In generating-function or bijective language, this construction corresponds to iterated applications of the CVS bijection; at each stage, a quadrangulation is folded to a smaller tree with associated matchings (Lionni et al., 2019).

3. Convergence and Scaling Limits

Rescaling parameters are fixed by the scaling of trees: each contour $\mathbf{C}_n^{(j)}$ is normalized by $\alpha_n^{(j)}\sim (2n)^{1/2^j}$ and each label $\mathbf{L}_n^{(j)}$ by $\beta_n^{(j)}\sim (2n)^{1/2^{j+1}}$, so $\mathbf{T}_n^{(D)}$ "lives" at scale $n^{1/2^D}$. Convergence of the normalized "snakes" is established in the space of pointed snakes ($\mathrm{PS}^D$), i.e., up to cyclical re-rootings of the canonical tours, with convergence in law under the appropriate quotient operation.

The main limit theorem [(Lionni et al., 2019), Thm 4.8] states:

$$\mathbf{r}_n^{\bullet}[D]\ \to\ \mathbf{r}^{\bullet}[D]$$

in distribution within the pointed-snake topology and subsequent feuilletage construction. The strategy is inductive in $D$, relying on tightness via Kolmogorov's Hölder estimates, finite-dimensional convergence via conditioned CLT for the branching random walk, and passing to equivalence classes to handle the lack of rerooting continuity. However, this topology does not immediately guarantee Gromov–Hausdorff convergence for the metric spaces, pending further analytical development.

4. Metric Invariants: Diameter and Hausdorff Dimension

A sharp probabilistic upper bound for the graph diameter is shown:

$$\mathrm{diam}\big(\mathbf{r}_n[D]\big)\leq O_p\big(n^{1/2^D}\big),$$

i.e., for every $\epsilon >0$ there exists $C$ such that $$\Pr(\operatorname{diam}(\mathbf{r}_n[D]) > C n^{1/2^D}) < \epsilon$$ [(Lionni et al., 2019), Rem 4.12]. In the scaling limit, the conjectural Hausdorff dimension is $2^D$, justified by the Hölder regularity of $h^{(D)}$ (upper bound), volume growth heuristics (lower bound), and explicit enumeration exponents:

$d_H(\mathbf{r}[D]) = 2^D.$

This formula recovers the known results for $D=1$ (CRT, $d_H=2$), $D=2$ (Brownian map, $d_H=4$), and posits $d_H=8$ for $D=3$.

5. Numerical Validation and Simulation Methodology

Extensive Monte Carlo simulations of $\mathbf{R}_n[D]$ for $D=2,3$ employ volume and distance-scaling analyses to estimate $d_H$ (Castro et al., 6 Nov 2025). The central observable is the distribution of rescaled pairwise distances:

$$n^{-1/d_H} r_n \rightsquigarrow \text{limiting law}$$

with $d_H$ extracted from least-squares alignment of histograms across system sizes and systematic exploration of histogram deciles (75%, 50%, 25%) to assess bias.

Algorithmically, each realization proceeds with generation of a Dyck path and label arrays for $\mathbf{T}_n^{(1)}$, iterative conjugation and relabeling for higher $j$, root-grafting arrays to encode equivalence classes, and final graph assembly via adjacency relations and identifications. For each realization, single-source distances from 10 randomly chosen roots are measured using breadth-first search, aggregated over $10^5$ realizations.

Simulation parameters vary by $D$ and geometry:

• For trees $\mathbf{T}_n^D$: $n$ up to $2^{26}$ ($D=3$)
• For maps/feuilletages $\mathbf{R}_n[D]$: $n$ up to $2^{27}$ ($D=3$)
• Fit parameters (e.g., $k_n$, $a$, $b$, $\delta$) capture finite-size corrections.

Empirically, for $D=2$, measured $d_H$ for both trees and quadrangulations falls within 3–5% of $4$, with statistical errors below $0.2$. For $D=3$, tree data cluster tightly around $d_H\approx 8.0$; feuilletage data exhibit finite-size drift with $d_H$ approaching $7.3$ at the largest volumes, compatible with $8$ after extrapolation and correction for known sampling biases. The main bias source is root selection: uniform sampling among tree corners rather than final vertices favors higher-corner-degree vertices, skewing distributions toward shorter radii and hence underestimating $d_H$ at finite $n$.

6. Special Cases and Universality

Specific instances of $\mathbf{r}[D]$ are:

• $D=0$: the unit circle $S^1$ ($\mathbf{r}[0]$)
• $D=1$: Aldous’ continuum random tree (CRT) ($\mathbf{r}[1]$)
• $D=2$: the Brownian map ($\mathbf{r}[2]$)
• $D\geq 3$: candidate high-dimensional analogues, not corresponding to known mated-CRT or Liouville quantum gravity universality classes.

For $D=3$, the measured string-susceptibility exponent is $\gamma_s = -3/2$, but there is no mated-CRT map that reproduces both string susceptibility and $d_H=8$. This suggests that the $D=3$ feuilletage may define a genuinely new universality class of random geometry (Castro et al., 6 Nov 2025).

7. Open Directions and Future Prospects

Two immediate technical directions are advancing simulation capability to larger system sizes and eliminating systematic bias via unbiased vertex-rooting or multi-root distance averaging. A central theoretical question is the extension of convergence results to the Gromov–Hausdorff topology. There is also significant interest in scaling these investigations to $D=4$ and beyond, to determine the persistence of $d_H=2^D$ and further characterize the universality class distinctions indicated by the present data. The $D$-random feuilletages thus present a promising and concrete route to probing higher-dimensional random geometry in a combinatorially well-controlled, background-independent framework (Lionni et al., 2019, Castro et al., 6 Nov 2025).