  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 $\lambda=1$.
