  We prove necessary and sufficient conditions for lattice Schr\"{o}dinger
operators to have a zero energy bound state in arbitrary dimension. The two
criteria are sharp, complementary, and depend crucially on both the dimension
and asymptotic behaviour of the potential. The method relies on a discrete
variant of Agmon's comparison principle which is also proven. Our results
represent a discrete variant of the recent criteria obtained in the continuous
setting by D. Hundertmark, M. Jex, and M. Lange [$\textit{Forum Mathematics,
Sigma}$ $\textbf{11}$ (2023)].
