### Abstract

We prove tail triviality of determinantal point processes μ on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine _{2}, Airy _{2}, Bessel _{2}, and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.

Original language | English |
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Pages (from-to) | 421-435 |

Number of pages | 15 |

Journal | Journal of Statistical Physics |

Volume | 170 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Discrete Approximations of Determinantal Point Processes on Continuous Spaces : Tree Representations and Tail Triviality.** / Osada, Hirofumi; Osada, Shota.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 170, no. 2, pp. 421-435. https://doi.org/10.1007/s10955-017-1928-2

}

TY - JOUR

T1 - Discrete Approximations of Determinantal Point Processes on Continuous Spaces

T2 - Tree Representations and Tail Triviality

AU - Osada, Hirofumi

AU - Osada, Shota

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We prove tail triviality of determinantal point processes μ on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine 2, Airy 2, Bessel 2, and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.

AB - We prove tail triviality of determinantal point processes μ on continuous spaces. Tail triviality has been proved for such processes only on discrete spaces, and hence we have generalized the result to continuous spaces. To do this, we construct tree representations, that is, discrete approximations of determinantal point processes enjoying a determinantal structure. There are many interesting examples of determinantal point processes on continuous spaces such as zero points of the hyperbolic Gaussian analytic function with Bergman kernel, and the thermodynamic limit of eigenvalues of Gaussian random matrices for Sine 2, Airy 2, Bessel 2, and Ginibre point processes. Our main theorem proves all these point processes are tail trivial.

UR - http://www.scopus.com/inward/record.url?scp=85035115601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035115601&partnerID=8YFLogxK

U2 - 10.1007/s10955-017-1928-2

DO - 10.1007/s10955-017-1928-2

M3 - Article

AN - SCOPUS:85035115601

VL - 170

SP - 421

EP - 435

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -