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We consider lattice triangulations, i.e., triangulations of the integer points in a polygon in Euclidean plane. 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 (the local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). By contrast, for \lambda>1 we show that the mixing time is exponential.

For thin rectangular regions we obtain sharp mixing time bounds for all values of \lambda<1. This is joint work with F. Martinelli, A. Sinclair and A. Stauffer