Probability Theory and Related Fields

, Volume 165, Issue 3–4, pp 725–770 | Cite as

Absolute continuity and singularity of Palm measures of the Ginibre point process

Article

Abstract

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).

Keywords

Ginibre point process Palm measure Absolute continuity Singularity 

Mathematics Subject Classification

60G30 60G55 82B21 60B20 60K35 

References

  1. 1.
    Chatterjee, S., Peled, R., Peres, Y., Romik, D.: Gravitational allocation to Poisson points. Ann. Math. 172, 617–671 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ginibre, J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ghosh, S.: Rigidity and tolerance in Gaussian zeroes and Ginibre eigenvalues: quantitative estimates. arXiv:1211.3506v1
  4. 4.
    Ghosh, S.: Determinantal processes and completeness of random exponentials: the critical case. arXiv:1211.2435v1
  5. 5.
    Ghosh, S., Peres, Y.: Rigidity and tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues. arXiv:1211.2381v2
  6. 6.
    Heicklen, D., Lyons, R.: Change intolerance in spanning forests. J. Theor. Probab. 16(1), 47–58 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Holroyd, A.E., Soo, T.: Insertion and deletion tolerance of point processes. Electron. J. Probab. 18(74), 1–24 (2013)Google Scholar
  8. 8.
    Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, vol. 51. AMS Univ. Lect. Ser. (2009)Google Scholar
  9. 9.
    Kakutani, S.: On equivalence of infinite product measures. Ann. Math. 49, 214–224 (1948)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kostlan, E.: On the spectra of Gaussian matrices. Linear Algebra Appl. 162–164, 385–388 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  12. 12.
    Osada, H.: Infinite-dimensional stochastic differential equations related to random matrices. Probab. Theory Relat. Fields 153(3–4), 471–509 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Osada, H.: Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials. Ann. Probab. 41, 1–49 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Osada, H., Shirai, T.: Variance of the linear statistics of the Ginibre random point field. RIMS Kôkyûroku Bessatsu B6, 193–200 (2008)MathSciNetMATHGoogle Scholar
  15. 15.
    Rider, B., Virág, B.: Complex determinantal processes and \(H^1\) noise. Elecron. J. Probab. 12(45), 1238–1257 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Skorohod, A .V.: On the differentiability of measures which correspond to stochastic processes I. Processes with independent increments. Teor. Verojat. Primen. 2, 418–444 (1957)MathSciNetGoogle Scholar
  19. 19.
    Shirai, T., Takahashi, Y.: Random point fields associated with certain Fredholm determinants I: Fermion, Poisson and Boson processes. J. Funct. Anal. 205, 414–463 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Shirai, T.: Large deviations for the Fermion point process associated with the exponential kernel. J. Stat. Phys. 123, 615–629 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Takahashi, Y.: Absolute continuity of Poisson random fields. Publ. Res. Inst. Math. Sci. 26(4), 629–647 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan

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