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 , this establishes the existence of a dynamical phase transition for thin rectangles with critical point at λ = 1.
Caputo, P., Martinelli, F., Sinclair, A., Stauffer, A. (2016). Dynamics of lattice triangulations on thin rectangles. ELECTRONIC JOURNAL OF PROBABILITY, 21(29), 1-22 [10.1214/16-EJP4321].