Linear systems and determinantal random point fields

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 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−(2x+s+t)AB 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 . 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.