Linear systems and determinantal random point fields

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 $L^2(0,\mathrm{\infty })$ 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 $\mathrm{\Gamma }$ with kernel ${\varphi }_{(x)}(s+t)=C{e}^{-(2x+s+t)A}B$ such that $\det (I+(z-1)\Gamma\Gamma^\dagger)$ is the generating function of a determinantal random point field.