## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 25, Number 3 (2015), 1650-1685.

### Random lattice triangulations: Structure and algorithms

Pietro Caputo, Fabio Martinelli, Alistair Sinclair, and Alexandre Stauffer

#### Abstract

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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 25, Number 3 (2015), 1650-1685.

**Dates**

First available in Project Euclid: 23 March 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1427124139

**Digital Object Identifier**

doi:10.1214/14-AAP1033

**Mathematical Reviews number (MathSciNet)**

MR3325284

**Zentralblatt MATH identifier**

1293.52011

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 68W20: Randomized algorithms 05C81: Random walks on graphs

**Keywords**

Triangulations spatial mixing Glauber dynamics mixing times rapid mixing

#### Citation

Caputo, Pietro; Martinelli, Fabio; Sinclair, Alistair; Stauffer, Alexandre. Random lattice triangulations: Structure and algorithms. Ann. Appl. Probab. 25 (2015), no. 3, 1650--1685. doi:10.1214/14-AAP1033. https://projecteuclid.org/euclid.aoap/1427124139