Completely Monotonic Fredholm Determinants

Abstract

A functionf(x) is called completely monotonic if (−1)mf(m)(x) > 0. In random matrix theory when the associated orthogonal polynomials have Freud weights, it is known that the expectation of having m eigenvalues of a random Hermitian matrix in an interval is a multiple of (−1)m times the m-th derivative of a Fredholm determinant at λ = 1. In this work we extend these results in two directions: (1) from λ = 1 to λ ∈ (−∞, 1] for general orthogonal weights; (2) from matrices to trace class operators. We also provide many special function examples that are Fredholm determinants of trace class operators in disguise.