### Abstract

We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda^{|\sigma|}$ where $\lambda>0$ is a parameter and $|\sigma|$ denotes the total edge length of the triangulation. When $\lambda\in(0,1)$ and $k$ is fixed, we prove a tight upper bound of order $n^2$ for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order $\exp(\Omega(n^2))$ for $\lambda>1$ [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at $\lambda=1$.

Original language | English |
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Pages (from-to) | 1-22 |

Journal | Electronic Journal of Probability |

Volume | 21 |

DOIs | |

Publication status | Published - 14 Apr 2016 |

### Keywords

- math.PR
- cs.DM
- math.CO

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## Cite this

Caputo, P., Martinelli, F., Sinclair, A., & Stauffer, A. (2016). Dynamics of lattice triangulations on thin rectangles.

*Electronic Journal of Probability*,*21*, 1-22. https://doi.org/10.1214/16-EJP4321