Blower, Gordon (2009) *Linear systems and determinantal random point fields.* Journal of Mathematical Analysis and Applications, 355 (1). pp. 311-334. ISSN 0022-247X

## Abstract

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.

Item Type: | Article |
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Journal or Publication Title: | Journal of Mathematical Analysis and Applications |

Additional Information: | The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 355 (1), 2009, © ELSEVIER. |

Uncontrolled Keywords: | Determinantal point processes ; Random matrices ; Inverse scattering |

Subjects: | Q Science > QA Mathematics |

Departments: | Faculty of Science and Technology > Mathematics and Statistics |

ID Code: | 11330 |

Deposited By: | Professor Gordon Blower |

Deposited On: | 13 Aug 2008 14:41 |

Refereed?: | Yes |

Published?: | Published |

Last Modified: | 17 Dec 2017 01:36 |

Identification Number: | |

URI: | http://eprints.lancs.ac.uk/id/eprint/11330 |

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