Random Lattice Triangulations
summary
Defines lattice triangulations on the n×n integer grid, puts the Gibbs weight mu(σ) ∝ λ^{total L1 edge length}, and uses Glauber flips (diagonal flip of a convex unit quadrilateral) to sample. Proves for λ<1 sufficiently small there is a pointwise Lyapunov function α^{|e|} crossing-sum that is a supermartingale, implying rapid convergence bounds, exponential tails for edge lengths, tightness of local measures, and robustness under arbitrary fixed constraints and polygonal domains. Empirically λ>1 shows macroscopic aligned regions and slow (likely exponential) mixing; λ=1 remains open.
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