Pietro Caputo, Fabio Martinelli, Alistair Sinclair, Alexandre Stauffer
The paper concerns lattice triangulations, i.e., 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 sigma has weight lambda^{|sigma|}, where lambda is a positive real parameter and |sigma| is the total length of the edges in sigma. Empirically, this model exhibits a "phase transition" at lambda=1 (corresponding to the uniform distribution): for lambda<1 distant edges behave essentially independently, while for lambda>1 very large regions of aligned edges appear. We substantiate this picture as follows. For lambda<1 sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for lambda>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.
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http://arxiv.org/abs/1211.1784
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