Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $\sigma$ has weight $\lambda^{|\sigma|}$, where $\lambda$ is a positive real parameter, and $|\sigma|$ is the total length of the edges in $\sigma$. Empirically, this model exhibits a "phase transition" at $\lambda=1$ (corresponding to the uniform distribution): for $\lambda<1$ distant edges behave essentially independently, while for $\lambda>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $\lambda<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $\lambda>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Growing uniform planar maps face by face

We provide “growth schemes” for inductively generating uniform random -angulations of the sphere with faces, as well as uniform random simple triangulations of the sphere with faces. In the case of -angulations, we provide a way to insert a new face at a random location in a uniform -angulation with faces in such a way that the new map is precisely a uniform -angulation with faces. Similarly, given a uniform simple triangulation of the sphere with faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer

The skeleton of the UIPT, seen from infinity

We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton "seen from infinity" of the UIPT and relate it to a simple Galton--Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the $2$-point function formula for random triangulations in the scaling limit due to Ambj{\o}rn and Watabiki.Comment: 34 pages, 14 figure

Statistics of planar graphs viewed from a vertex: A study via labeled trees

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges and/or vertices, as well as explicit values for the corresponding moments.Comment: 36 pages, 15 figures, tex, harvmac, eps

Uniform infinite planar triangulation and related time-reversed critical branching process

We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e. for a boundary component separating the ball from infinity). We show also that outside of R-ball a contour exists that has length linear in R.Comment: 27 pages, 5 figures, LaTe

Critical and Multicritical Semi-Random (1+d)-Dimensional Lattices and Hard Objects in d Dimensions

We investigate models of (1+d)-D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)-D Lorentzian surfaces, with fractal dimensions $d_F=k+1$, k=1,2,3,..., as well as a new model of (1+2)-D critical tetrahedral complexes, with fractal dimension $d_F=12/5$. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1+d)-D Lorentzian lattices and directed-site lattice animals in (1+d) dimensions.Comment: 44 pages, 15 figures, tex, harvmac, epsf, references adde

A note on weak convergence results for uniform infinite causal triangulations

We discuss uniform infinite causal triangulations and equivalence to the size biased branching process measure - the critical Galton-Watson branching process distribution conditioned on non-extinction. Using known results from the theory of branching processes, this relation is used to prove weak convergence of the joint length-area process of a uniform infinite causal triangulations to a limiting diffusion. The diffusion equation enables us to determine the physical Hamiltonian and Green's function from the Feynman-Kac procedure, providing us with a mathematical rigorous proof of certain scaling limits of causal dynamical triangulations.Comment: 23 pages, 2 figure