Journal of the Mathematical Society of Japan

Ginibre-type point processes and their asymptotic behavior

Tomoyuki SHIRAI

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Abstract

We introduce Ginibre-type point processes as determinantal point processes associated with the eigenspaces corresponding to the so-called Landau levels. The Ginibre point process, originally defined as the limiting point process of eigenvalues of the Ginibre complex Gaussian random matrix, can be understood as a special case of Ginibre-type point processes. For these point processes, we investigate the asymptotic behavior of the variance of the number of points inside a growing disk. We also investigate the asymptotic behavior of the conditional expectation of the number of points inside an annulus given that there are no points inside another annulus.

Article information

Source
J. Math. Soc. Japan Volume 67, Number 2 (2015), 763-787.

Dates
First available in Project Euclid: 21 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1429624603

Digital Object Identifier
doi:10.2969/jmsj/06720763

Mathematical Reviews number (MathSciNet)
MR3340195

Zentralblatt MATH identifier
1319.60102

Subjects
Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

Keywords
Ginibre point process determinantal point processes Landau Hamiltonian reproducing kernel Bargmann–Fock space Laguerre polynomial

Citation

SHIRAI, Tomoyuki. Ginibre-type point processes and their asymptotic behavior. J. Math. Soc. Japan 67 (2015), no. 2, 763--787. doi:10.2969/jmsj/06720763. https://projecteuclid.org/euclid.jmsj/1429624603.


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References

  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure. Appl. Math., 14 (1961), 187–214.
  • G. Chistyakov, Yu. Lyubarskii and L. Pastur, On completeness of random exponentials in the Bargmann–Fock space, J. Math. Phys., 42 (2001), 3754–3768.
  • O. Costin and J. L. Lebowitz, Gaussian fluctuation in random matrices, Phys. Rev. Lett., 75 (1995), 69–72.
  • J.-D. Deuschel and D. W. Stroock, Large Deviations, Pure. Appl. math., 137, Academic Press, Boston, 1989.
  • J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys., 6 (1965), 440–449.
  • A. Haimi and H. Hedenmalm, The polyanalytic Ginibre ensembles, J. Stat. Phys., 153 (2013), 10–47.
  • J. B. Hough, Large deviations for the zero set of an analytic function with diffusing coefficients..
  • J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Determinantal processes and independence, Probab. Surv., 3 (2006), 206–229.
  • T. Hupfer, M. Leschke and S. Warzel, Upper bounds on the density of states of single Landau levels broadened by Gaussian random potentials, J. Math. Phys., 42 (2001), 5626–5641.
  • B. Jancovici, J. L. Lebowitz and G. Manificat, Large charge fluctuations in classical Coulomb systems, J. Statist. Phys., 72 (1993), 773–787.
  • M. Krishnapur, Overcrowding estimates for zeroes of planar and hyperbolic Gaussian analytic functions, J. Stat. Phys., 124 (2006), 1399–1423.
  • N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972.
  • A. Nishry, Asymptotics of the hole probability for zeros of random entire functions, Int. Math. Res. Not. IMRN., 2010, no.,15 (2010), 2925–2946.
  • H. Osada and T. Shirai, Variance of the linear statistics of the Ginibre random point field, RIMS Kôkyûroku Bessatsu, B6 (2008), 193–200.
  • Y. Peres and B. Virág, Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Math., 194 (2005), 1–35.
  • N. Rohringer, J. Burgdörfer and N. Macris, Bargmann representation for Landau levels in two dimensions, J. Phys. A, 36 (2003), 4173–4190.
  • T. Shirai, Large deviations for the fermion point process associated with the exponential kernel, J. Stat. Phys., 123 (2006), 615–629.
  • T. Shirai, Limit theorems for random analytic functions and their zeros, RIMS Kôkyûroku Bessatsu, B34 (2012), 335–359.
  • T. Shirai and Y. Takahashi, Random point fields associated with certain Fredholm determinants. I: Fermion, Poisson and boson point processes, J. Funct. Anal., 205 (2003), 414–463.
  • M. Sodin and B. Tsirelson, Random complex zeroes. III. Decay of the hole probability, Israel J. Math., 147 (2005), 371–379.
  • A. Soshnikov, Determinantal random point fields, Russian Math. Surveys, 55 (2000), 923–975.
  • A. Soshnikov, Gaussian limit for determinantal random point fields, Ann. Probab., 30 (2002), 171–187.