Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the in- teger points in a polygon in R2 whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation σ has weight λ|σ|, where λ is a positive real parameter, and |σ| is the total length of the edges in σ. Empirically, this model exhibits a “phase transition” at λ = 1 (correspond- ing to the uniform distribution): for λ 1 very large regions of aligned edges appear. We substantiate this picture as follows. For λ 1 we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations