Abstract
We study some properties of hyperbolic Gaussian analytic functions of intensity L in the unit ball of ℂn. First we deal with the asymptotics of fluctuations of linear statistics as L → ∞. Then we estimate the probability of large deviations (with respect to the expected value) of such linear statistics and use this estimate to prove a hole theorem.
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J. Buckley, Random zero sets of analytic functions and traces of functions in Fock spaces, Ph.D. Thesis, Universitat de Barcelona, Barcelona, 2013.
J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, Vol. 51, American Mathematical Society, Providence, RI, 2009.
W. Rudin, Function Theory in the Unit Ball of ℂn, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
B. Shiffman and S. Zelditch, Number variance of random zeros on complex manifolds. II: smooth statistics, Pure and Applied Mathematics Quarterly 6 (2010), 1145–1167.
B. Shiffman, S. Zelditch and S. Zrebiec, Overcrowding and hole probabilities for random zeros on complex manifolds, Indiana University Mathematics Journal 57 (2008), 1977–1997.
M. Sodin, Zeros of Gaussian analytic functions, Mathematical Research Letters 7 (2000), 371–381.
M. Sodin and B. Tsirelson, Random complex zeroes. I. Asymptotic normality, Israel Journal of Mathematics 144 (2004), 125–149.
M. Sodin and B. Tsirelson Random complex zeroes. III. Decay of the hole probability, Israel Journal of Mathematics 147 (2005), 371–379.
M. Stoll, Invariant Potential Theory in the Unit Ball of C n, London Mathematical Society Lecture Note Series, Vol. 199, Cambridge University Press, Cambridge, 1994.
S. Zrebiec, The zeros of flat Gaussian random holomorphic functions on ℂn, and hole probability, Michigan Mathematical Journal 55 (2007), 269–284.
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Authors supported by the Generalitat de Catalunya (grant 2014 SGR 00289) and the Spanish Ministerio de Economía y Competividad (projects MTM2011-27932-C02-01).
The first author is also supported by the Raymond and Beverly Sackler Post-Doctoral Scholarship.
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Buckley, J., Massaneda, X. & Pridhnani, B. Gaussian analytic functions in the unit ball. Isr. J. Math. 209, 855–881 (2015). https://doi.org/10.1007/s11856-015-1239-8
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DOI: https://doi.org/10.1007/s11856-015-1239-8