Israel Journal of Mathematics

, Volume 152, Issue 1, pp 105–124 | Cite as

Random complex zeroes, II. Perturbed lattice



We show that the flat chaotic analytic zero points (i.e. zeroes of a random entire function\(\psi (z) = \sum {_{k = 0}^\infty \zeta } k\frac{{z^k }}{{\sqrt {k!} }}\) where ζ0, ζ1, … are independent standard complex-valued Gaussian variables) can be regarded as a random perturbation of a lattice in the plane. The distribution of the distances between the zeroes and the corresponding lattice points is shift-invariant and has a Gaussian-type decay of the tails.


Entire Function Poisson Point Process Main Lemma Stable Marriage Vietoris Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Aharoni,Infinite matching theory, Discrete Mathematics95 (1991), 5–22.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    M. Ajtai, J. Komlós and G. Tusnády,On optimal matchings, Combinatorica4 (1984), 259–264.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P. Bleher, B. Shiffman and S. Zelditch,Poincaré-Lelong approach to universality and scaling of correlations between zeros, Communications in Mathematical Physics208 (2000), 771–785.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    N. Dunford and J. Schwartz,Linear Operators. Part I. General Theory, Interscience, New York, 1958.Google Scholar
  5. [5]
    J. H. Hannay,Chaotic analytic zero points: exact statistics for those of a random spin state, Journal of Physics. A. Mathematical and General29 (1996), L101-L105.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    C. Hoffman, A. E. Holroyd and Y. Peres,A stable marriage of Poisson and Lebesgue, arXiv:math.PR/0505668.Google Scholar
  7. [7]
    A. E. Holroyd and Y. Peres,Extra heads and invariant allocations, Annals of Probability33 (2005), 31–52.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    L. Hörmander,The Analysis of Linear Partial Differential Operators, Vol. I, Distribution Theory and Fourier Analysis, Springer-Verlag, Berlin, 1983.Google Scholar
  9. [9]
    J. Moser,On the volume elements on a manifold, Transactions of the American Mathematical Society120 (1965), 286–294.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Y. Peres and B. Virag,Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Mathematica, to appear. arXiv:math.PR/0310297.Google Scholar
  11. [11]
    M. Sodin and B. Tsirelson,Random complex zeroes, I. Asymptotic normality, Israel Journal of Mathematics144 (2004), 125–149.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    M. Sodin and B. Tsirelson,Random complex zeroes, III. Decay of the hole probability, Israel Journal of Mathematics147 (2005), 371–379.MATHMathSciNetGoogle Scholar
  13. [13]
    S. M. Srivastava,A Course on Borel Sets, Springer-Verlag, Berlin, 1998.MATHGoogle Scholar
  14. [14]
    M. Talagrand,Matching theorems and empirical discrepancy computations using majorizing measures, Journal of the American Mathematical Society7 (1994), 455–537.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  1. 1.School of MathematicsTel Aviv UniversityTel AvivIsrael

Personalised recommendations