## Electronic Journal of Probability

### Dynamics of lattice triangulations on thin rectangles

#### 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 $\lambda =1$.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 29, 22 pp.

Dates
Accepted: 4 April 2016
First available in Project Euclid: 14 April 2016

https://projecteuclid.org/euclid.ejp/1460652930

Digital Object Identifier
doi:10.1214/16-EJP4321

Mathematical Reviews number (MathSciNet)
MR3492933

Zentralblatt MATH identifier
1336.60183

#### Citation

Caputo, Pietro; Martinelli, Fabio; Sinclair, Alistair; Stauffer, Alexandre. Dynamics of lattice triangulations on thin rectangles. Electron. J. Probab. 21 (2016), paper no. 29, 22 pp. doi:10.1214/16-EJP4321. https://projecteuclid.org/euclid.ejp/1460652930

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