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

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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.

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Notes

  1. On \(\mathbb {R}^D\), the largest tuple of points \((x_1,\cdots ,x_k)\) such that \(i\ne j\Rightarrow d(x_i,x_j)=1\) is bounded by \((D+1)\) when it is not the case on maps.

  2. This is likely the case for some models of random 4-dimensional triangulations when using the distance on the dual graph [8, 50], and is conjectured for another model of random 3-dimensional triangulations using the distance in the triangulation.

  3. More precisely, this last case corresponds to families of random \(D\)-dimensional triangulations whose diameter is small or a.s. bounded when the number of simplices goes to infinity, for which a scaling limit cannot be defined.

  4. We use encodings of planar trees by non-crossing partitions whose Kreweras complements are matchings (disjoint sets encode all the vertices of the tree). For more details, see the end of Sect 6.1.

  5. More precisely, this is the case when starting from a uniform rooted pointed quadrangulation with n faces.

  6. The asymptotic dependence of their diameters in the number of edges.

  7. “Spatial” is an adjective that is used to distinguish the different processes into play: prosaically, it is just a usual linear Brownian motion.

  8. A “canonical” discrete curvature [63] is defined on a \(D\)-dimensional triangulation by assuming that all edges have the same length. Then, the discrete curvature locally depends only on the number of \(D\)-simplices around each \((D-2)\)-simplex.

  9. If these random discrete space-times converge towards a certain continuum limit (scaling limit), finding out if general relativity is recovered as an effective theory in a certain “non-quantum” limit could involve defining suitable observables on the scaling limit, that would converge to their classical (i.e. non-quantum) values throughout a coarse-graining process, or knowing how to describe this continuum limit in a field theoretic way, and then renormalizing this theory to translate it to our scales. But there are no known spaces so far to serve as toy-models for addressing this question.

  10. All results discussed here are in the Euclidean case, in which time is not considered. Introducing “time” can be done by requiring some additional causality condition on the \(D\)-dimensional triangulations [3, 51]. Numerical simulations seem to indicate that the continuum limit in dimension 4 has promising properties, however no exact result exists so far.

  11. The normalized Brownian excursion can be obtained by rescaling the excursion of the Brownian motion which straddles 1. From here, it can be seen that the trajectory of the Brownian excursion inherits from the Brownian motion of many features. For example, it has a countable number of local minima or maxima (Revuz-Yor [64, Chap. III, 3.26]). Besides, the Brownian motion has the strong Markov property, and the property that the set \(Z=\{t:B_t=0\}\) is a.s. a closed set without isolated point (see [64, Chap. III Prop 3.12]), allows seeing that a.s., 0 (or any other point \(x\in [0,1]\)) is not a local maximum or minimum: for any \(x\in [0,1]\), one has a.s. \(\inf {\dot{x}}=\sup {\dot{x}}\), even if a.s. x is not a local maximum. Hence, in \( T_{\mathsf{e}}\) the set of leaves has a.s. Lebesgue measure 1, and \(T_{\mathsf{e}}\) is a.s. a binary tree.

  12. When one quotients a topological space as we did when we introduced \(\sim _{ D}\), it may happen that the “projected distance” is not a distance: for example, let \(E= [0,1]\) equipped with the usual distance |.|, and quotiented by the equivalence relation \(x\sim y\) iff \(x=y\) or \(x,y \in {\mathbb {Q}}\) (identify rational numbers). Clearly, the quotient space \(E^{\star }\) is not reduced to a single point, but \(d^{\star }(x,y)=0\) for all \(x,y\in E^{\star }\) under “the inherited distance \(d^\star \)”. Hence, \(d^\star \) is not a distance, since \(d^\star (x,y)=0 \not \Rightarrow x=y\). If one further quotients \(E^{\star }\) by \(x\sim ^{\star } y\) when \(d^\star (x,y)=0\), then the space \(E^{\star }\) becomes trivial, reduced to a single point.

  13. To avoid this problem, it would suffice to prove that a.s., \(\#{{\,\mathrm{argmin}\,}}(\varvec{\ell }^{(j)})=1\), for the iterated process \(\varvec{\ell }^{(j)}\).

  14. Between integer points, \(\mathbf{L}_n\) is linear, and it is folklore and easy to check, that the tightness can be proved by proving the moment condition only at these discretization points.

  15. The common denomination would be labeled map, but we chose indexed to avoid confusion with the labeling of the trees and the label processes in the rest of the text.

  16. For trees, we rather consider rooted trees together with a canonical indexing of the half-edges, as detailed below.

  17. This is not the convention used for processes in Sect. 3, in which the corners of the trees are labeled from 0 to \(2n-1\) going around the tree clockwise. This should be kept in mind when comparing the various examples. Apart from that, one then recovers the contour and height sequences of the tree, as defined in Sect. 3.

  18. Planar trees with n edges are also mapped bijectively to non-crossing partitions on n ordered elements. To obtain the description in the text from this last one, the non-crossing partition on n elements is completed by its Kreweras complement \(C \rightarrow C\sqcup {{\overline{C}}}\). In the context of this paper, the permutations we obtain naturally have matchings as Kreweras complements.

  19. The min argmin is a corner of \(T_{n}^{(1)}\), and if e is the edge added from this corner towards the pointed vertex \(\nu \), \({\tau }_n^{(2)}\) has to be rerooted on the corner of \(\nu \) which precedes e on \(\nu \) counterclockwise in order to obtain \(T_{n}^{(2)}\) (the orange arrow in Fig. 19).

  20. The cycles of \(\sigma ^{(1)}\) induce a non-crossing partition on \(C_2=\{0<\ldots <4n-1\}\), but it is \((\sigma ^{(1)})^{-1}\) that respects the ordering of \(C_2\), since when going around \(T_{n}^{(1)}\) counterclockwise, we go around \({\tau }_n^{(2)}\) clockwise

  21. The reason why we don’t have to shift the first element of the set \(C_j\) to obtain the various iterated trees is precisely because the root of \({\tau }_n^{(j+1)}\) is chosen as dual to the root of \({\tau }_n^{(j)}\).

  22. Note that there is a bijection [9, 10, 32, 50] between orientable colored triangulations labeled this way and so-called \((D+1)\)-constellations [12] (but without any assumption on the genus of the constellation).

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Acknowledgements

This works has been partially supported by ANR GRAAL (ANR-14-CE25-0014). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. ERC-2016-STG 716083, “CombiTop”). LL thanks Dario Benedetti and Valentin Bonzom for useful discussions on the physical motivations.

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  1. CNRS UMR 8243 - IRIF, Université Paris Diderot - Paris 7, Bâtiment Sophie Germain, 75205, Paris Cedex 13, France

    Luca Lionni

  2. CNRS, UMR 5800 - LaBRI, Univ. Bordeaux, Bordeaux INP, 351 cours de la Libération, 33405, Talence Cedex, France

    Jean-François Marckert

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Appendix

Appendix

1.1 Folding a Planar Map to Get a 3D Object?

This informal section is devoted to explaining that a construction similar to that of \(R_N[3]\) allow constructing “3D discrete objects”, for a certain notion of 3D explained below.

Fig. 23
figure 23

Folding a planar map to get a 3D object

Full size image

First, consider the graph \(G_n\) having as vertex set the points of the discrete cube \(\{1,...,n\}^3\), and as edges, the pairs of points at Euclidean distance 1 from each other. In the sense above, \(G_n\) is a “3D-spherical-like” graph, as it is the 1-skeleton of a gluing of cubes which discretizes a 3-ball. And more generally, it may be argued that any reasonable definition of spherical graph dimension should give dimension 3 or higher to a graph having \(G_n\) as subgraph. Now, we will present \(G_n\) as a refolded map, very similar to some elements of \(R_N(3)\) for some N.

The idea is the following: consider the chessboard type figure represented in Picture 1, Fig. 23, obtained by taking a section of \(\mathbb {R}^2\) (or \(\mathbb {Z}^2\)), in which a unit square out of two has been removed. Each of the lacking square is then used in Picture 2 as the basis of a cube with the same lacking face: half of these cubes are placed above the plane, half below in such a way that the west-south corners of the cubes above (resp. below) forms, up to some translation, a section of \((2\mathbb {Z})^2\). The obtained object is a quadrangulation (with a big square boundary) since all its inner faces have degree 4. Now, change a bit of point of view in Picture 2, and view what is represented as a kind of fabric with a texture: the fabric being the plane, the texture being made of cubes above and below the fabric. Now, imagine a large piece of fabric which is folded and sewn as on Picture 3: it is a quadrangulation with 4 “layers”, each of them made by two large strips of fabric “at distance 2”. Each layer is also at distance 2 from the next layer. Here, ”distance 2” has to be understood for the usual metric in \(\mathbb {R}^3\) since clearly this picture can be embedded isometrically in \(\mathbb {R}^3\). Now add the texture to the fabric! Picture 4 figures what happens inside one layer (between two strips that are face to face) or between two layers of the fabric (outside the big quadrangulations, between the two strips that are face to face). Up to some details concerning the parity of the strips, the cubes that have size 1 and which are placed on “planes” at distance 2, intersect at their vertices only. Observe again Picture 4, and imagine the \(\mathbb {Z}^3\) lattice in between the two planes: notice that every edge of the lattice belongs to exactly one of the cubes. What is shown on Picture 4 represents what happens inside each of the 4 layers, and between 3 inter-layers of Picture 3, so, it concerns a total section of width 13 of \(\mathbb {Z}^3\).

Hence, it is possible to fold and identify vertices in quadrangulations to construct some objects whose underlying graphs contain \(G_n\).

As a matter of fact, it may be argued that this example is not in the set \(\mathbf{R}_{N}[3]\) for any N large (one of the reasons being that the extracted trees \((t_2,t_3)\) of the textured fabric with the Cori-Vauquelin-Schaeffer bijection provides a tree \(t_2\) whose height process has not the required increments \(+1, -1\) or 0). But we hope that it illustrates the fact that such “3D-like objects” can arise in the construction we propose.

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Lionni, L., Marckert, JF. Iterated Foldings of Discrete Spaces and Their Limits: Candidates for the Role of Brownian Map in Higher Dimensions. Math Phys Anal Geom 24, 39 (2021). https://doi.org/10.1007/s11040-021-09410-5

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