The Markovian hyperbolic triangulation of Curien and Werner (2013) is the unique (in law) random tiling of the hyperbolic disk into ideal hyperbolic triangles, with corners on the boundary at infinity, whose law is invariant under all Mobius transformations of the disk and which satisfies a spatial Markov property: the parts of the triangulation on the two sides of an edge are conditionally independent. Its existing construction is indirect, based on Poisson point processes with infinite, sigma-finite intensity measures, and the authors closed their paper with three open questions asking for a more direct understanding of this object and of its possible dynamics: (1) an alternative construction that explains "directly" why the triangulation exists; (2) natural discrete models, such as percolation interfaces in random triangulations, that give rise to it in the scaling limit; and (3) natural Mobius-invariant dynamics on the space of triangulations that leave the law invariant, with fast decorrelation of distant triangles. None of the three questions has been resolved or even partially answered as of 2026; all three remain open.

Background

The hyperbolic plane can be represented by the open unit disk D\mathbb{D} endowed with the Poincare metric (the distance function giving the disk constant negative curvature); in this model, the orientation-preserving isometries of the hyperbolic plane are exactly the Mobius transformations of the disk, that is, the fractional-linear conformal bijections of D\mathbb{D} onto itself, which form a three-dimensional Lie group identifiable with PSL(2,R)\mathrm{PSL}(2,\mathbb{R}). An ideal triangle is a hyperbolic triangle whose three vertices lie on the boundary circle ∂D\partial\mathbb{D}, "at infinity"; all ideal triangles are congruent, and a triangulation of D\mathbb{D} into ideal triangles is determined by which triples of boundary points are joined.

Curien and Werner asked whether there exists a random such triangulation combining two properties well known from different parts of probability theory. The first is invariance in law under all Mobius transformations of D\mathbb{D}, the exact hyperbolic counterpart of the conformal invariance that characterizes the Schramm-Loewner evolution (SLE), the family of random curves arising as scaling limits of interfaces of critical lattice models. The second is a spatial Markov property: conditionally on a triangle of the triangulation, the restrictions of the triangulation to the complementary domains are independent, so that the edges of the triangulation play the role that interfaces play in a nearest-neighbor model from statistical mechanics. Their main theorem states that there exists exactly one law of a complete (that is, covering) Mobius-invariant triangulation of D\mathbb{D} with this spatial Markov property; this is the Markovian hyperbolic triangulation.

Both the uniqueness argument and the existence proof are indirect. The conditional law of the third vertex of the triangle adjacent to a given edge would have to be invariant under all Mobius transformations fixing the two endpoints of that edge, and every such measure has infinite mass concentrated near the endpoints; it is a multiple of the image of the measure ξ(dx)=dx/x\xi(dx)=dx/x on the positive half-line under a boundary-fixing map. The construction therefore proceeds through Poisson point processes (random countable point sets with independent scattering) with such infinite, sigma-finite intensities, in the spirit of the theory of Levy processes (random processes with stationary independent increments). The resulting object is striking: almost surely no two triangles of the triangulation are adjacent, with infinitely many ever-thinner (in the Euclidean sense) triangles separating any two of them, and the closure of the union of all triangle boundaries has Hausdorff dimension equal to one, making the triangulation much "thinner" than other random triangulations of the disk, such as Aldous's uniform triangulation (the scaling limit of uniformly random triangulations of an nn-gon) or the recursive triangulations of the disk studied by Curien and Le Gall.

The paper closes with three open questions: the first two ask for a direct and robust explanation of why this object exists, and the third asks what natural dynamics preserve its law. It would be desirable to have an alternative construction, for instance via an auxiliary Poissonian model, an allocation idea (a rule assigning regions of space to points of a random point configuration), or statistical-physics arguments, that would explain "directly" why the triangulation exists. It is natural to ask for discrete models, such as percolation interfaces in random triangulations, that would give rise to the Markovian hyperbolic triangulation in the scaling limit, just as the discrete percolation model "explains" special properties of SLE6\mathrm{SLE}_6; the authors note that Omer Angel's exploration of percolation interfaces on the uniform infinite planar triangulation (UIPT), the infinite random planar map obtained as the local limit of large uniform finite triangulations and explored by revealing its faces one at a time in a "peeling" procedure, has a reminiscent Markovian structure but gives rise to a different scaling limit. And it is natural to ask whether there exist Mobius-invariant dynamics on the space of triangulations that leave the law invariant, for instance a continuous evolution in which each triangle follows a Brownian motion in the three-dimensional Lie group of Mobius transformations, with the evolutions of distant triangles decorrelating fast.

These questions sit between two research lines that have remained largely separate. On the discrete side, the combinatorial theory of Markovian random planar maps, including the classification of half-planar maps by peeling invariance (Angel and Ray) and Curien's stochastic hyperbolic triangulations Tκ\mathbb{T}_\kappa, which are Markovian in the peeling sense and hyperbolic in the sense of exponential volume growth and positive random-walk speed (Angel, Nachmias and Ray), contains high-genus local limit theorems (Budzinski and Louf) and a near-critical scaling limit of Brownian-map type called the hyperbolic Brownian plane (Budzinski); here 'Brownian-map type' refers to the Brownian map, the canonical scaling limit of large random planar maps viewed as random metric surfaces. These, however, are combinatorial graphs: they are not tilings of the hyperbolic plane by ideal triangles, they are not Mobius-invariant, and in the hyperbolic regime they admit no scaling limit at all as random metric spaces. On the continuum side, isometry-invariant random tessellations of the hyperbolic plane have been constructed, namely right-angled hexagon tilings via statistical-mechanics-style arguments (Kenyon), the hyperbolic Poisson-Voronoi and Poisson-Delaunay tessellations (Benjamini, Paquette and Pfeffer), and the ideal Poisson-Voronoi tessellation obtained as the low-intensity limit of hyperbolic Poisson-Voronoi tessellations (d'Achille, Curien, Enriquez, Lyons and Unel); but none possesses the spatial Markov property, and none has been connected to the Markovian hyperbolic triangulation. The three open questions ask precisely for the missing links.

Problem Statement

The problem is the block of three open questions posed by Curien and Werner at the end of The Markovian hyperbolic triangulation (Section 6), which ask, beyond the original indirect construction, for a direct and robust understanding of the unique Mobius-invariant Markovian triangulation of the hyperbolic disk: an alternative construction explaining its existence, discrete models giving rise to it in the scaling limit, and Mobius-invariant dynamics preserving its law. Stated faithfully, the questions are:

  1. Find an alternative construction of the Markovian Mobius-invariant triangulation that explains "directly" why it exists.

  2. Identify natural discrete models, for example percolation interfaces in random triangulations in the spirit of Angel's exploration of percolation interfaces on the uniform infinite planar triangulation (which however gives rise to a different scaling limit), that give rise to such Markovian hyperbolic triangulations in the scaling limit, thereby providing another heuristic or rigorous justification of the existence of the triangulation, in the way the discrete percolation model "explains" the locality and reversibility properties of SLE6\mathrm{SLE}_6.

  3. Determine what natural Mobius-invariant dynamics (if they exist) on the set of triangulations leave the measure invariant; in particular, whether there is a "continuous" evolution, in some appropriate topology, under which the evolutions of triangles that are far apart de-correlate fast with this distance.

The three questions are posed independently by the authors, and a solution of the problem consists of a rigorous theorem that resolves at least one of them: an explicit alternative construction with the properties requested in (1); a specified discrete model together with a proved convergence to the Markovian hyperbolic triangulation as requested in (2); or, as requested in (3), a precisely defined natural Mobius-invariant dynamics that provably leaves the Curien-Werner law invariant and under which the evolutions of distant triangles de-correlate fast with distance. For (3), the rigid flow generated by the defining Mobius symmetries does not by itself qualify, and the existence question may also be resolved negatively by proving that no such law-preserving dynamics exist. Results about neighboring objects, such as combinatorial Markovian triangulations and their Brownian-map-type scaling limits, hyperbolic Poisson-Voronoi and Poisson-Delaunay tessellations, ideal Poisson-Voronoi tessellations, or isometry-invariant hexagon tilings, do not by themselves answer the questions unless the link to the Curien-Werner triangulation itself is established.

References

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