We consider random lattice triangulations of n×k rectangular regions
with weight λ∣σ∣ where λ>0 is a parameter and
∣σ∣ denotes the total edge length of the triangulation. When
λ∈(0,1) and k is fixed, we prove a tight upper bound of order n2
for the mixing time of the edge-flip Glauber dynamics. Combined with the
previously known lower bound of order exp(Ω(n2)) for λ>1 [3],
this establishes the existence of a dynamical phase transition for thin
rectangles with critical point at λ=1