Gaussian Limit for Determinantal Random Point Fields by Alexander Soshnikov

We prove that, under fairly general conditions, a properly rescaled de-terminantal random point field converges to a generalized Gaussian random process. 1. Introduction and formulation of results. Let E be a locally compact Hausdorff space satisfying the second axiom of countability, B—σ-algebra of Borel subsets and µ a σ-finite measure on (E, B), such that µ(K) < ∞ for any compact K ⊂ E. We denote by X the space of locally finite configurations of particles in E: X = {ξ = (x i) ∞ i=−∞ : x i ∈ E ∀i, and for any compact K ⊂ E # K (ξ) := #(x i : x i ∈ K) < +∞}. A σ-algebra F of measurable subsets of X is generated by the cylinder sets C B n = {ξ ∈ X : # B (ξ) = n}, where B is a Borel set with a compact closure and n ∈ Z 1 + = {0, 1, 2,. . .}. Let P be a probability measure on (X, F). A triple (X, F , P) is called a random point field (process) (see [4, 17–19]). In this paper we will be interested in a special class of random point fields called deter-minantal random point fields. It should be noted that most, if not all the important examples of determinantal point fields arise when E = k i=1 E i (here we use the notation for the disjoint union), E i ∼ = R d or Z d and µ is either the Lebegue or the counting measure. We will, however, develop our results in the general setting (our arguments will not require significant changes). Let dx i , i = 1,. .. , n, be disjoint infinitesimally small subsets around the x i 's. Suppose that a probability to find a particle in each dx i (with no restrictions outside of n i=1 dx i) is proportional to