## The Annals of Applied Probability

### Random lattice triangulations: Structure and algorithms

#### 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

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

Digital Object Identifier
doi:10.1214/14-AAP1033

Mathematical Reviews number (MathSciNet)
MR3325284

Zentralblatt MATH identifier
1293.52011

#### 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

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