Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Linear systems and determinantal random point fields.
AU - Blower, Gordon
N1 - The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 355 (1), 2009, © ELSEVIER.
PY - 2009/7
Y1 - 2009/7
N2 - Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from systems of differential equations with rational coefficient. More generally, this paper considers symmetric Hamiltonian systems and determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives sufficient condition for a self-adjoint operator to be the Hankel operator on L^2(0, \infinity ) from a linear system in continuous time; so this paper expresses certain kernels as squares of Hankel operators. For suitable systems (-A,B,C) with one dimensional input and output spaces, there exists a Hankel operator \Gamma with kernel \phi such that \det (I+(z-1)\Gamma\Gamma^\dagger) is the generating function of a dterminnatal random point field on (0, \infty ). The invrse scattering transform for the Zhakarov--Shabat system involves an Gelfand--Levitan integral equation such that the daigonal of te solution gives the derivative of th log generating function. Some dtererminatal print fields in random matrix theory satisfy similar results.
AB - Tracy and Widom showed that fundamentally important kernels in random matrix theory arise from systems of differential equations with rational coefficient. More generally, this paper considers symmetric Hamiltonian systems and determines the properties of kernels that arise from them. The inverse spectral problem for self-adjoint Hankel operators gives sufficient condition for a self-adjoint operator to be the Hankel operator on L^2(0, \infinity ) from a linear system in continuous time; so this paper expresses certain kernels as squares of Hankel operators. For suitable systems (-A,B,C) with one dimensional input and output spaces, there exists a Hankel operator \Gamma with kernel \phi such that \det (I+(z-1)\Gamma\Gamma^\dagger) is the generating function of a dterminnatal random point field on (0, \infty ). The invrse scattering transform for the Zhakarov--Shabat system involves an Gelfand--Levitan integral equation such that the daigonal of te solution gives the derivative of th log generating function. Some dtererminatal print fields in random matrix theory satisfy similar results.
KW - Determinantal point processes
KW - Random matrices
KW - Inverse scattering
U2 - 10.1016/j.jmaa.2009.01.070
DO - 10.1016/j.jmaa.2009.01.070
M3 - Journal article
VL - 355
SP - 311
EP - 334
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
SN - 0022-247X
IS - 1
ER -