Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions
DOI:
https://doi.org/10.15407/mag12.03.205Анотація
Вивчено асимптотичнi закони для числа i просторового розподiлу компонент зв'язностi нульових множин гладких гаусiвських випадкових функцiй кiлькох дiйсних змiнних. Основними прикладами є рiзнi гаусiвськi ансамблi дiйсних полiномiв (алгебраїчних i тригонометричних) високого степеня на сферi i на торi, а також трансляцiйно-iнварiантнi гладкi гаусiвськi випадковi функцiї в евклiдовому просторi, що звуженi на великi областi.
Mathematical Subject Classification: 60G15.
Ключові слова:
гладкі гаусiвські випадкові функцiї кiлькох дiйсних змiнних, число компонент зв'язностi нульової множини, ергодичністьПосилання
M.E. Becker, Multiparameter Groups of Measure-Preserving Tranformations: a Simple Proof of Wiener’s Ergodic Theorem. — Ann. Prob. 9 (1981), 504–509. https://doi.org/10.1214/aop/1176994423
D. Beliaev and Z. Kereta, On Bogomolny–Schmit Conjecture. — J. Phys. A: Math. Theor. 46 (2013), 45503. arXiv:1310.2747 https://doi.org/10.1088/1751-8113/46/45/455003
S. Bochner, Vorlesungen Uber Fouriersche Integrale. Leipzig, 1932.
V. Bogachev, Gaussian Measures. Mathematical Surveys and Monographs 62 AMS, Providence, RI, 1998. https://doi.org/10.1090/surv/062
E. Bogomolny and C. Schmit, Percolation Model for Nodal Domains of Chaotic Wave Functions. — Phys. Rev. Lett. 88 (2002), 114102. arXiv:nlin/0110019v1; Random Wavefunctions and Percolation, J. Phys. A 40 (2007), 14033–14043. arXiv:0708.4335v1 https://doi.org/10.1088/1751-8113/40/47/001
J. Bourgain, On Toral Eigenfunctions and the Random Wave Model. — Israel J. Math. 201 (2014), 611–630. arXiv:1303.2881.
J. Bourgain and Z. Rudnick, Restriction of Toral Eigenfunctions to Hypersurfaces and Nodal Sets. — Geom. Funct. Anal. 22 (2012), 878–937. arXiv:1105.0018v1. https://doi.org/10.1007/s00039-012-0186-3
Y. Canzani and P. Sarnak, On the Topology of the Zero Sets of Monochromatic Random Waves. arXiv:1412.4437.
Y. Fyodorov, A. Lerario, and E. Lundberg, On the Number of Connected Components of Random Algebraic Hypersurfaces. — J. Geom. Phys. 95 (2015), 1–20. arXiv:1404.5349. https://doi.org/10.1016/j.geomphys.2015.04.006
D. Gayet and J.-Y. Welschinger, Lower Estimates for the Expected Betti Numbers of Random Real Hypersurfaces. — J. London Math. Soc. (2) 90 (2014), 105–120. arXiv:1303.3035 https://doi.org/10.1112/jlms/jdu018
D. Gayet and J.-Y. Welschinger, Expected Topology of Random Real Algebraic Submanifolds. J. Inst. Math. Jussieu (2015). arXiv:1307.5287.
D. Gayet and J.-Y. Welschinger, Universal Components of Random Zero Sets. arXiv:1503.01582.
U. Grenander, Stochastic Processes and Statistical Inference. — Ark. Mat. 1 (1950), 195–277. https://doi.org/10.1007/BF02590638
S. Janson, Gaussian Hilbert Spaces. Cambridge Univ. Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511526169
M. Hairer, An Introduction to Stochastic PDEs. arXiv:0907.4178
Y. Katznelson, An Introduction to Harmonic Analysis. Cambridge Univ. Press, Cambridge, 2004. https://doi.org/10.1017/CBO9781139165372
H. Komatsu, Prof. Yosida’s Proof of the Plancherel and the Bochner Theorems for Locally Compact Abelian Groups. — J. Math. Kyoto Univ. 41 (2001), 739–747.
K. Konrad, Asymptotic Statistics of Nodal Domains of Quantum Chaotic Billiards in the Semiclassical Limit. Senior Thesis, Dartmouth College, 2012.
E. Kostlan, On the Expected Number of Real Roots of a System of Random Polynomial Equations. Foundations of Computational Mathematics (Hong Kong, 2000), 149–188, World Sci. Publ., River Edge, New York, 2002.
P. Kurlberg and I. Wigman, Non-universality of the Nazarov–Sodin Constant. — C. R. Math. Acad. Sci. Paris 353 (2015), 101–104. arXiv:1406.7449 https://doi.org/10.1016/j.crma.2014.09.026
H.J. Landau and L.A. Shepp, On the Supremum of a Gaussian Process. — Sankhy, Ser. A 32 (1970), 369–378.
A. Lerario and E. Lundberg, Statistics on Hilbert’s Sixteenth Problem. Int. Math. Res. Notices (2015). arXiv:1212.3823
T.L. Malevich, Contours That Arise When the Zero Level is Crossed by Gaussian Fields. — Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 16 (1972), 20–23.
M. Nastasescu, The Number of Ovals of a Real Plane Curve. Senior Thesis, Princeton, 2011.
F. Nazarov and M. Sodin, On the Number of Nodal Domains of Random Spherical Harmonics. — Amer. J. Math. 131 (2009), 1337–1357. arXiv:0706.2409v1 https://doi.org/10.1353/ajm.0.0070
Y. Rozenshein, The Number of Nodal Components of Arithmetic Random Waves. Master Thesis, Tel Aviv University, 2015. arXiv:1604.00638
P. Sarnak, Letter to B. Gross and J. Harris on Ovals of Random Plane Curves, May 2011. http://publications.ias.edu/sarnak/paper/510
P. Sarnak and I. Wigman, Topologies of Nodal Sets of Random Band Limited Functions. arXiv:1312.7858
M. Sodin, Lectures on Random Nodal Portraits. In: Probability and Statistical Physics in St. Petersburg. Proc. Symp. Pure Math. 91. Amer. Math. Soc., Providencs RI, 2016. Available online http://www.math.tau.ac.il/ sodin/SPB-Lecture-Notes.pdf
B. Tsirelson, Gaussian Geasures. Lecture Notes. Tel Aviv University, Fall 2010.http://www.tau.ac.il/~tsirel/Courses/Gauss3/main.html
