Gibbsianness of Fermion Random Point Fields

We consider fermion (or determinantal) random point fields on Euclidean space R. Given a bounded, translation invariant, and positive definite integral operator J on L(R), we introduce a determinantal interaction for a system of particles moving on R as follows: the n points located at x1, · · · , xn ∈ R have the potential energy given by U (x1, · · · , xn) := − log det(j(xi − xj))1≤i,j≤n, where j(x − y) is the integral kernel function of the operator J . We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d ≥ 2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I + J) is a Gibbs measure admitted to the specification.