Random lattice triangulations: structure and algorithms
Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre O. Stauffer · The University of Bath Online Publications Store (The University of Bath) · 2013
The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in R 2 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 the structure and dynamics of random lattice triangulations.