Abstract

We consider lattice triangulations as triangulations of the integer points in the square [0; n]x[0; n]. Our focus is on random triangulations in which the probability of obtaining a given lattice triangulation T is proportional to \lambda^|T|, where \lambda is a positive real parameter and |T| is the total length of the edges in T. 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 defined), 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). By contrast, for \lambda > 1 we show that the mixing time is exponential. Joint work with Pietro Caputo, Fabio Martinelli and Alistair Sinclair.