  We study the well-known variational and large deviation principle for graph
homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method
to deduce those principles under minimal a priori assumptions. The only
ingredient specific to the model is a discrete Kirszbraun theorem i.e. an
extension theorem for graph homomorphisms. All other ingredients are of a
general nature not specific to the model. They include elementary
combinatorics, the compactness of Lipschitz functions and a simplicial
Rademacher theorem. Compared to the literature, our proof does not need any
other preliminary results like e.g. concentration or strict convexity of the
local surface tension. Therefore, the method is very robust and extends to more
complex and subtle models, as e.g. the homogenization of limit shapes or
graph-homomorphisms to a regular tree.

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