The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in @math 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 @math has weight @math , where @math is a positive real parameter, and @math is the total length of the edges in @math . Empirically, this model exhibits a "phase transition" at @math (corresponding to the uniform distribution): for @math very large regions of aligned edges appear. We substantiate this picture as follows. For @math 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.
We mention finally that our mixing time result for the special case of 1-dimensional regions with @math is related to work of Greenberg, Pascoe and Randall @cite_21 on lattice paths. Those authors use a similar path coupling argument, but for a different probability distribution on lattice paths: in their model paths are biased according to the area under the path, while in ours the bias depends on the excursions of the path from a fixed line.
@cite_21
1544652657
Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z2, and for bias λ > d2 in Zd, when d > 2. Moreover, our bounds on the mixing time are optimal on d-dimensional hyper-cubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.