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 $\lambda =1$.