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Random Witten Laplacians on infinite coverings of compact manifolds are considered. The probabilistic representations of the corresponding heat kernels are given. The finiteness of the von Neumann traces of the corresponding semigroups is proved, and the short-time asymptotics of the corresponding super-trace is computed. Examples associated with Gibbs measures on configuration spaces and product manifolds are considered.

Random Witten Laplacians: traces of semigroups, $L^2$-Betti numbers and index