Random lattice triangulations
Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre O. Stauffer · 2013
The paper concerns lattice triangulations, i.e., triangulations of the integer 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 (corresponding to the uniform distribution): for λ1 very large regions of aligned edges appear. We substantiate this picture as follows. For λ1 we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on spatial mixing properties and dynamics of random lattice triangulations.