  We analyze a general class of self-adjoint difference operators
$H_\varepsilon = T_\varepsilon + V_\varepsilon$ on
$\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well
potential and $\varepsilon$ is a small parameter. We construct a Finslerian
distance $d$ induced by $H_\varepsilon$ and show that short integral curves are
geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with
a rate controlled by the Finsler distance to the well. This is analog to
semiclassical Agmon estimates for Schr\"odinger operators.
