Linear systems and determinantal random point fields

TL;DR

This paper explores the connection between linear Hamiltonian systems, Hankel operators, and determinantal point fields, extending the understanding of kernels in random matrix theory through spectral and operator analysis.

Contribution

It characterizes kernels from Hamiltonian systems as squares of Hankel operators and links them to determinantal point fields via inverse spectral problems.

Findings

Kernels from Hamiltonian systems can be expressed as Hankel operators.

Determinantal point fields can be generated from specific Hankel operators.

Provides conditions under which operators correspond to linear systems in continuous time.

Abstract

Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from differential equations with rational coefficients. More generally, this paper considers symmetric Hamiltonian systems abd determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives a sufficient condition for a self-adjoint operator to be the Hankel operator on $L^2(0, \infty)$ from a linear system in continuous time; thus this paper expresses certain kernels as squares of Hankel operators. For a suitable linear system $(-A,B,C)$ 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 $\det (I+(z-1)\Gamma\Gamma^\dagger)$ is the generating function of a determinantal random point field.