Gibbsianness of fermion random point fields

Abstract

We consider fermion (or determinantal) random point fields on Euclidean space $\mbR^d$. Given a bounded, translation invariant, and positive definite integral operator $J$ on $L^2(\mbR^d)$, we introduce a determinantal interaction for a system of particles moving on $\mbR^d$ as follows: the $n$ points located at $x_1,...,x_n\in \mbR^d$ have the potential energy given by $$ U^{(J)}(x_1,...,x_n):=-\log\det(j(x_i-x_j))_{1\le i,j\le n}, $$ 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 $d\ge 2$ 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.