We analyze a general class of self-adjoint difference operators \(H_\varepsilon = T_\varepsilon + V_\varepsilon\,\, {\rm on}\,\, \ell^2((\varepsilon {\mathbb{Z}})^d)\), where V
ε is a one-well potential and ε is a small parameter. We construct a Finslerian distance d induced by H
ε 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 Schrödinger operators.