# Dynamics of lattice triangulations on thin rectangles

Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer

Research output: Contribution to journalArticlepeer-review

5 Citations (SciVal)

## 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

• math.PR
• cs.DM
• math.CO

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