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



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 \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).


Ginibre point process Palm measure Absolute continuity Singularity 

Mathematics Subject Classification

60G30 60G55 82B21 60B20 60K35 



The authors would like to thank the anonymous referee for the careful reading and helpful comments which improve our manuscript. The first author (HO)’s work was supported in part by JSPS Grant-in-Aid for Scientific Research (A) No. 24244010 and (B) No. 21340031. The second author (TS)’s work was supported in part by JSPS Grant-in-Aid for Scientific Research (B) No. 22340020 and (B) No. 26287019.


  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

Personalised recommendations