Traces of semigroups associated with interacting particle systems (Q886128)

The authors start from a complete connected, oriented manifold of infinite volume with a lower bounded curvature. They also assume that an infinite discrete group of isometries on \(X\) is given such that \(X/G\) is a compact Riemannian manifold. The set \( \Gamma _{X}\) of all locally finite subsets of \(X\) is called the configuration space of \(X\) and is assumed to be equipped with a Gibbs measure which is invariant under the natural action of \(G\) on the configuration space and satisfies a condition known under the name of a ``Ruelle bound''. Finally, they consider a special random measure on \(X\). In this context, they then study the Witten Laplacian on \(X\). In particular, they prove the finiteness of the associated \(\theta\)-function, and its von Neumann realization, and the existence of the corresponding \(L ^{2}\)-Betti numbers.