Spectral asymptotics of the Witten Laplacian on the lattice

A discrete analogue of the Witten Laplacian on the n‐dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low‐lying spectrum. Our proof, inspired by work of Helffer‐Klein‐Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of some metastable Markov processes. These naturally arise in Statistical mechanics in the context of disordered mean field models, like the Random field Curie‐Weiss model.