  We study random triangulations of the integer points $[0,n]^2
\cap\mathbb{Z}^2$, where each triangulation $\sigma$ has probability measure
$\lambda^{|\sigma|}$ with $|\sigma|$ denoting the sum of the length of the
edges in $\sigma$. Such triangulations are called \emph{lattice
triangulations}. We construct a height function on lattice triangulations and
prove that, in the whole subcritical regime $\lambda<1$, the function behaves
as a \emph{Lyapunov function} with respect to Glauber dynamics; that is, the
function is a supermartingale. We show the applicability of the above result by
establishing several features of lattice triangulations, such as tightness of
local measures, exponential tail of edge lengths, crossings of small triangles,
and decay of correlations in thin rectangles. These are the first results on
lattice triangulations that are valid in the whole subcritical regime
$\lambda<1$. In a very recent work with Caputo, Martinelli and Sinclair, we
apply this Lyapunov function to establish tight bounds on the mixing time of
Glauber dynamics in thin rectangles that hold for all $\lambda<1$. The Lyapunov
function result here holds in great generality; it holds for triangulations of
general lattice polygons (instead of the $[0,n]^2$ square) and also in the
presence of arbitrary constraint edges.
