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 $\lambda=1$