### Abstract

Original language | English |
---|---|

Pages (from-to) | 1-22 |

Journal | Electronic Journal of Probability |

Volume | 21 |

DOIs | |

Publication status | Published - 14 Apr 2016 |

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### Keywords

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

### Cite this

*Electronic Journal of Probability*,

*21*, 1-22. https://doi.org/10.1214/16-EJP4321

**Dynamics of lattice triangulations on thin rectangles.** / Caputo, Pietro; Martinelli, Fabio; Sinclair, Alistair; Stauffer, Alexandre.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Dynamics of lattice triangulations on thin rectangles

AU - Caputo, Pietro

AU - Martinelli, Fabio

AU - Sinclair, Alistair

AU - Stauffer, Alexandre

PY - 2016/4/14

Y1 - 2016/4/14

N2 - 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 $\lambda=1$.

AB - 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 $\lambda=1$.

KW - math.PR

KW - cs.DM

KW - math.CO

UR - http://dx.doi.org/10.1214/16-EJP4321

UR - http://dx.doi.org/10.1214/16-EJP4321

U2 - 10.1214/16-EJP4321

DO - 10.1214/16-EJP4321

M3 - Article

VL - 21

SP - 1

EP - 22

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -