  The paper concerns lattice triangulations, that is, triangulations of the
integer points in a polygon in $\mathbb{R}^2$ 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.
