  We develop a new robust technique to deduce variance principles for
non-integrable discrete systems. To illustrate this technique, we show the
existence of a variational principle for graph homomorphisms from $\Z^m$ to a
$d$-regular tree. This seems to be the first non-trivial example of a
variational principle in a non-integrable model. Instead of relying on
integrability, the technique is based on a discrete Kirszbraun theorem and a
concentration inequality obtained through the dynamic of the model. As a
consequence of this result, we obtain the existence of a continuum of
shift-invariant ergodic gradient Gibbs measures for graph homomorphisms from
$\Z^m$ to a regular tree.
