TY - JOUR

T1 - The Jancovici-Lebowitz-Manificat law for large fluctuations of random complex zeroes

AU - Nazarov, F.

AU - Sodin, M.

AU - Volberg, A.

N1 - Funding Information:
Partially supported by the National Science Foundation, DMS grant 0501067.
Funding Information:
Partially supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grants 357/04 and 171/07.

PY - 2008/12

Y1 - 2008/12

N2 - Consider a Gaussian Entire Function f(z) = ∑k=0 ∞ζkzk/√k!, where ζ0, ζ1,... are Gaussian i.i.d. complex random variables. The zero set of this function is distribution invariant with respect to the isometries of the complex plane. Let n(R) be the number of zeroes of f in the disk of radius R. It is easy to see that double-struck En(R) = R2, and it is known that the variance of n(R) grows linearly with R (Forrester and Honner). We prove that, for every α > 1/2, the tail probability ℙ {|n(R) - R 2| > Rα} behaves as exp [-R φ(α)] with some explicit piecewise linear function φ(α). For some special values of the parameter α, this law was found earlier by Sodin and Tsirelson, and by Krishnapur. In the context of charge fluctuations of a one-component Coulomb system of particles of one sign embedded into a uniform background of another sign, a similar law was discovered some time ago by Jancovici, Lebowitz and Manificat.

AB - Consider a Gaussian Entire Function f(z) = ∑k=0 ∞ζkzk/√k!, where ζ0, ζ1,... are Gaussian i.i.d. complex random variables. The zero set of this function is distribution invariant with respect to the isometries of the complex plane. Let n(R) be the number of zeroes of f in the disk of radius R. It is easy to see that double-struck En(R) = R2, and it is known that the variance of n(R) grows linearly with R (Forrester and Honner). We prove that, for every α > 1/2, the tail probability ℙ {|n(R) - R 2| > Rα} behaves as exp [-R φ(α)] with some explicit piecewise linear function φ(α). For some special values of the parameter α, this law was found earlier by Sodin and Tsirelson, and by Krishnapur. In the context of charge fluctuations of a one-component Coulomb system of particles of one sign embedded into a uniform background of another sign, a similar law was discovered some time ago by Jancovici, Lebowitz and Manificat.

UR - http://www.scopus.com/inward/record.url?scp=55149087689&partnerID=8YFLogxK

U2 - 10.1007/s00220-008-0646-7

DO - 10.1007/s00220-008-0646-7

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AN - SCOPUS:55149087689

SN - 0010-3616

VL - 284

SP - 833

EP - 865

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -