Gibbsianness of fermion random point fields

Abstract

We consider fermion (or determinantal) random point fields on Euclidean space ℝ d . Given a bounded, translation invariant, and positive definite integral operator J on L2(ℝ d ), we introduce a determinantal interaction for a system of particles moving on ℝ d as follows: the n points located at x1,· · ·,x n ∈ ℝ d have the potential energy given by 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)−1 is a Gibbs measure admitted to the specification.