Representations of stack triangulations in the plane

Stack triangulations appear as natural objects when defining an increasing family of triangulations by successive additions of vertices. We consider two different probability distributions for such objects. We represent, or "draw" these random stack triangulations in the plane $\R^2$ and study the asymptotic properties of these drawings, viewed as random compact metric spaces. We also look at the occupation measure of the vertices, and show that for these two distributions it converges to some random limit measure.Comment: 29 pages, 13 figure

Planar maps, circle patterns and 2d gravity

Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point process. We show that this measure can be expressed as: (1) a sum over 3-spanning-trees partitions of the edges of the Delaunay triangulations; (2) the volume form of a K\"ahler metric over the space of Delaunay triangulations, whose prepotential has a simple formulation in term of ideal tessellations of the 3d hyperbolic space; (3) a discretized version (involving finite difference complex derivative operators) of Polyakov's conformal Fadeev-Popov determinant in 2d gravity; (4) a combination of Chern classes, thus also establishing a link with topological 2d gravity.Comment: Misprints corrected and a couple of footnotes added. 42 pages, 17 figure

Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in arXiv:1401.3297. The proof relies on a combinatorial argument and the Goulden--Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane $T$ is weakly Markovian in the sense that the probability to observe a finite triangulation $t$ around the root only depends on the perimeter and volume of $t$, then $T$ is a mixture of PSHT.Comment: 36 pages, 10 figure

Transversal structures on triangulations: a combinatorial study and straight-line drawings

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change

The Ising model on the random planar causal triangulation: bounds on the critical line and magnetization properties

We investigate a Gibbs (annealed) probability measure defined on Ising spin configurations on causal triangulations of the plane. We study the region where such measure can be defined and provide bounds on the boundary of this region (critical line). We prove that for any finite random triangulation the magnetization of the central spin is sensitive of the boundary conditions. Furthermore, we show that in the infinite volume limit, the magnetization of the central spin vanishes for values of the temperature high enough.Comment: 28 pages, 2 figures, 1 section adde