Linear systems and determinantal random point fields

Abstract Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from systems of differential equations with rational coefficients. More generally, this paper considers symmetric Hamiltonian systems and determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on L2(0,∞) from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For suitable linear systems (−A,B,C) with one dimensional input and output spaces, there exists a Hankel operator Γ with kernel φ(x)(s+ t) = Ce B such that gx(z) = det(I+(z−1)ΓΓ†) is the generating function of a determinantal random point field on (0,∞). The inverse scattering transform for the Zakharov–Shabat system involves a Gelfand–Levitan integral equation such that the trace of the diagonal of the solution gives ∂ ∂x log gx(z). Some determinantal point fields in random matrix theory satisfy similar results.