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Fluctuations of the increment of the argument for the Gaussian entire function. (English) Zbl 1373.60068

Summary: The Gaussian entire function is a random entire function, characterised by a certain invariance with respect to isometries of the plane. We study the fluctuations of the increment of the argument of the Gaussian entire function along planar curves. We introduce an inner product on finite formal linear combinations of curves (with real coefficients), that we call the signed length, which describes the limiting covariance of the increment. We also establish asymptotic normality of fluctuations.

MSC:

60G15 Gaussian processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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[1] Breuer, P., Major, P.: Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425-441 (1983). doi:10.1016/0047-259X(83)90019-2 · Zbl 0518.60023 · doi:10.1016/0047-259X(83)90019-2
[2] Dalmao, F., Nourdin, I., Peccati, G., Rossi, M.: Phase Singularities in Complex Arithmetic Random Waves (2016). arXiv:1608.05631 [math.PR] · Zbl 1467.60034
[3] Diaconis, P., Evans, Steven N.: Linear functionals of eigenvalues of random matrices. Trans. Am. Math. Soc. 353(7), 2615-2633 (2001). doi:10.1090/S0002-9947-01-02800-8 · Zbl 1008.15013 · doi:10.1090/S0002-9947-01-02800-8
[4] Feldheim, N.: Variance of the number of zeroes of shift-invariant Gaussian analytic functions (2015). arXiv:1309.2111 [math.PR] · Zbl 1221.30008
[5] Ghosh, S., Lebowitz, J.: Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey (2016). arXiv:1608.07496 [math.PR] · Zbl 1390.60104
[6] Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, vol. 51. American Mathematical Society, Providence (2009) · Zbl 1190.60038
[7] Hughes, CP; Keating, JP; O’Connell, N., No article title, Commun. Math. Phys., 220, 429-451 (2001) · Zbl 0987.60039 · doi:10.1007/s002200100453
[8] Hughes, C.P., Nikeghbali, A., Yor, M.: An arithmetic model for the total disorder process. Probab. Theory Relat. Fields 141(1-2), 47-59 (2008). doi:10.1007/s00440-007-0079-9 · Zbl 1144.60020 · doi:10.1007/s00440-007-0079-9
[9] Janson, S.: Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129. Cambridge University Press, Cambridge (1997) · Zbl 0887.60009 · doi:10.1017/CBO9780511526169
[10] Kahane, J.-P.: Some Random Series of Functions. Cambridge Studies in Advanced Mathematics, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1985) · Zbl 0571.60002
[11] Kang, N.-G., Makarov, N.G.: Gaussian free field and conformal field theory. Astérisque 353, viii+136 (2013) (English, with English and French summaries) · Zbl 1280.81004
[12] Joel, L.: Charge fluctuations in Coulomb systems. Phys. Rev. A 27(3), 1491-1494 (1983). doi:10.1103/PhysRevA.27.1491 · Zbl 0547.35086 · doi:10.1103/PhysRevA.27.1491
[13] Marinucci, D., Peccati, G., Rossi, M., Wigman, I.: Non-universality of nodal length distribution for arithmetic random waves. Geom. Funct. Anal. doi:10.1007/s00039-016-0376-5 · Zbl 1347.60013
[14] Montgomery, H.L.: Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1994) · Zbl 0814.11001
[15] Nazarov, F., Sodin, M.: Fluctuations in random complex zeroes: asymptotic normality revisited. Int. Math. Res. Not. IMRN 24, 5720-5759 (2011) · Zbl 1242.60051 · doi:10.1093/imrn/rnr007
[16] Nazarov, F., Sodin, M.: Correlation functions for random complex zeroes: strong clustering and local universality. Commun. Math. Phys. 310(1), 75-98 (2012). doi:10.1007/s00220-011-1397-4 · Zbl 1238.60059 · doi:10.1007/s00220-011-1397-4
[17] Nazarov, F., Sodin, M., Volberg, A.: The Jancovici-Lebowitz-Manificat law for large fluctuations of random complex zeroes. Commun. Math. Phys. 284(3), 833-865 (2008). doi:10.1007/s00220-008-0646-7 · Zbl 1221.30008 · doi:10.1007/s00220-008-0646-7
[18] Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177-193 (2005). doi:10.1214/009117904000000621 · Zbl 1097.60007 · doi:10.1214/009117904000000621
[19] Peccati, G., Tudor, C.A.: Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics, pp. 247-262. Springer, Berlin (1857) · Zbl 1063.60027
[20] Shiffman, B., Zelditch, S.: Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18(4), 1422-1475 (2008). doi:10.1007/s00039-008-0686-3 · Zbl 1168.32009 · doi:10.1007/s00039-008-0686-3
[21] Sodin, M., Tsirelson, B.: Random complex zeroes. I. Asymptotic normality. Israel J. Math. 144, 125-149 (2004). doi:10.1007/BF02984409 · Zbl 1072.60043 · doi:10.1007/BF02984409
[22] Tsirelson, B.: Moderate deviations for random fields and random complex zeroes (2008). arXiv:0801.1050 [math.PR] · Zbl 1144.60020
[23] Wieand, K.: Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123(2), 202-224 (2002). doi:10.1007/s004400100186 · Zbl 1044.15016 · doi:10.1007/s004400100186
[24] Wikipedia. Argument principle—Wikipedia, The Free Encyclopedia. https://en.wikipedia.org/w/index.php?title=Argument_principle&oldid=771098532 (2017). Accessed 26 April 2017
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