# Dynamics of lattice triangulations on thin rectangles

Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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 1-22 Electronic Journal of Probability 21 https://doi.org/10.1214/16-EJP4321 Published - 14 Apr 2016

### Fingerprint

Rectangle
Triangulation
Dynamical Phase Transition
Glauber Dynamics
Mixing Time
Flip
Critical point
Lower bound
Upper bound
Denote
Lower bounds
Phase transition

• math.PR
• cs.DM
• math.CO

### Cite this

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

In: Electronic Journal of Probability, Vol. 21, 14.04.2016, p. 1-22.

Research output: Contribution to journalArticle

Caputo, Pietro ; Martinelli, Fabio ; Sinclair, Alistair ; Stauffer, Alexandre. / Dynamics of lattice triangulations on thin rectangles. In: Electronic Journal of Probability. 2016 ; Vol. 21. pp. 1-22.
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