Dynamics of Lattice Triangulations on Thin Rectangles

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 n for the mixing time of the edge-flip Glauber dynamics. Combined with the previously known lower bound of order exp(Ω(n)) for λ > 1 [3], this establishes the existence of a dynamical phase transition for thin rectangles with critical point at λ = 1.