Linear systems and determinantal random point fields

Abstract

In random matrix theory, determinantal random point fields describe the distribution of eigenvalues of self-adjoint matrices from the generalized unitary ensemble. This paper considers symmetric Hamiltonian systems and determines the properties of kernels and associated determinantal random point fields that arise from them; this extends work of Tracy and Widom. The inverse spectral problem for self-adjoint Hankel operators gives sufficient conditions for a self-adjoint operator to be the Hankel operator on L 2 ( 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 ) = C e − ( 2 x + s + t ) A B such that g x ( z ) = det ( I + ( z − 1 ) Γ Γ † ) 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 g x ( z ) . When A ⩾ 0 is a finite matrix and B = C † , there exists a determinantal random point field such that the largest point has a generalised logistic distribution.