Israel Journal of Mathematics

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

Random complex zeroes, II. Perturbed lattice

  • Mikhail Sodin
  • Boris Tsirelson


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 
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  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

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