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 L(2)(0, infinity) 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 Gamma with kernel phi(x)(s + t) = Ce(-(2x+s+t)A) B Such that gx(z) = det(I + (z - 1)Gamma Gamma(dagger)) is the generating function of a determinantal random point field on (0, infinity). 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 partial derivative/partial derivative x log g(x)(z). When A >= 0 is a finite matrix and B = C(dagger), there exists a determinantal random point field such that the largest point has a generalised logistic distribution. (c) 2009 Elsevier Inc. All rights reserved.