Boundedness of the nodal domains of additive Gaussian fields
Author:
S. Muirhead
Journal:
Theor. Probability and Math. Statist. 106 (2022), 143-155
MSC (2020):
Primary 60G60; Secondary 60F99
DOI:
https://doi.org/10.1090/tpms/1169
Published electronically:
May 16, 2022
MathSciNet review:
4438448
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Abstract: We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets $\{f \le \ell \}$ of additive planar Gaussian fields are bounded almost surely at the critical level $\ell _c = 0$. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension $d \ge 3$ the excursion sets have unbounded components at all levels.
References
- Robert J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478
- Kenneth S. Alexander, Boundedness of level lines for two-dimensional random fields, Ann. Probab. 24 (1996), no. 4, 1653–1674. MR 1415224, DOI 10.1214/aop/1041903201
- Vincent Beffara, La percolation, et un résultat de S. Smirnov, Gaz. Math. 128 (2011), 5–14 (French). MR 2828326
- Sourav Chatterjee, Superconcentration and related topics, Springer Monographs in Mathematics, Springer, Cham, 2014. MR 3157205, DOI 10.1007/978-3-319-03886-5
- Robert C. Dalang and T. Mountford, Jordan curves in the level sets of additive Brownian motion, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3531–3545. MR 1837246, DOI 10.1090/S0002-9947-01-02811-2
- H. Duminil-Copin, A. Rivera, P.-F. Rodriguez, and H. Vanneuville, Existence of unbounded nodal hypersurface for smooth Gaussian fields in dimension $d \ge 3$, arXiv preprint arXiv:2108.08008 (2021).
- Nicolas Durrande, David Ginsbourger, and Olivier Roustant, Additive covariance kernels for high-dimensional Gaussian process modeling, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 3, 481–499 (English, with English and French summaries). MR 3076409, DOI 10.5802/afst.1342
- A. M. Dykhne, Conductivity of a two-dimensional two-phase system, Zh. Eksp. Teor. Fiz. 59 (1970), 110–115.
- A. Gandolfi, M. Keane, and L. Russo, On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation, Ann. Probab. 16 (1988), no. 3, 1147–1157. MR 942759, DOI 10.1214/aop/1176991681
- Christophe Garban and Hugo Vanneuville, Bargmann-Fock percolation is noise sensitive, Electron. J. Probab. 25 (2020), Paper No. 98, 20. MR 4136478, DOI 10.1214/20-ejp491
- Geoffrey Grimmett, Percolation, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR 1707339, DOI 10.1007/978-3-662-03981-6
- T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960), 13–20. MR 115221, DOI 10.1017/S0305004100034241
- M. B. Isichenko, Percolation, statistical topography, and transport in random media, Rev. Modern Phys. 64 (1992), no. 4, 961–1043. MR 1187940, DOI 10.1103/RevModPhys.64.961
- Harry Kesten, The critical probability of bond percolation on the square lattice equals ${1\over 2}$, Comm. Math. Phys. 74 (1980), no. 1, 41–59. MR 575895, DOI 10.1007/BF01197577
- M. Lidbetter, G. Lindgren, and Kh. Rot⋅sen, Èkstremumy sluchaĭnykh posledovatel′nosteĭ i protsessov, “Mir”, Moscow, 1989 (Russian). Translated from the English by V. P. Nosko; Translation edited and with a preface by Yu. K. Belyaev. MR 1004671
- S. A. Molchanov and A. K. Stepanov, Percolation in random fields. I, Teoret. Mat. Fiz. 55 (1983), no. 2, 246–256 (Russian, with English summary). MR 734878
- S. A. Molchanov and A. K. Stepanov, Percolation in random fields. I, Teoret. Mat. Fiz. 55 (1983), no. 2, 246–256 (Russian, with English summary). MR 734878
- S. Muirhead, A. Rivera, and H. Vanneuville (with an appendix by L. Köhler-Schindler), The phase transition for planar Gaussian percolation models without FKG, arXiv preprint arXiv:2010.11770 (2020).
- Stephen Muirhead and Hugo Vanneuville, The sharp phase transition for level set percolation of smooth planar Gaussian fields, Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 2, 1358–1390 (English, with English and French summaries). MR 4076787, DOI 10.1214/19-AIHP1006
- Gábor Pete, Corner percolation on $\Bbb Z^2$ and the square root of 17, Ann. Probab. 36 (2008), no. 5, 1711–1747. MR 2440921, DOI 10.1214/07-AOP373
- Alejandro Rivera, Talagrand’s inequality in planar Gaussian field percolation, Electron. J. Probab. 26 (2021), Paper No. 26, 25. MR 4235477, DOI 10.1214/21-EJP585
- Alejandro Rivera and Hugo Vanneuville, The critical threshold for Bargmann-Fock percolation, Ann. H. Lebesgue 3 (2020), 169–215 (English, with English and French summaries). MR 4060853, DOI 10.5802/ahl.29
- Kevin Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes, Statist. Probab. Lett. 106 (2015), 239–246. MR 3389997, DOI 10.1016/j.spl.2015.07.028
- R. Zallen and H. Scher, Percolation on a continuum and the localization-delocalization transition in amorphous semiconductors, Phys. Rev. B. 4 (1971), 4471–4479.
References
- R. J. Adler, An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 12, Institute of Mathematical Statistics, Hayward, CA, 1990. MR 1088478
- K. S. Alexander, Boundedness of level lines for two-dimensional random fields, Ann. Probab. 24 (1996), no. 4, 1653–1674. MR 1415224
- V. Beffara and D. Gayet, Percolation without FKG, arXiv preprint arXiv:1710.10644 (2017). MR 2828326
- S. Chatterjee, Superconcentration and related topics, Springer Monographs in Mathematics, Springer, Cham, 2014. MR 3157205
- R. C. Dalang and T. Mountford, Jordan curves in the level sets of additive Brownian motion, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3531–3545. MR 1837246
- H. Duminil-Copin, A. Rivera, P.-F. Rodriguez, and H. Vanneuville, Existence of unbounded nodal hypersurface for smooth Gaussian fields in dimension $d \ge 3$, arXiv preprint arXiv:2108.08008 (2021).
- N. Durrande, D. Ginsbourger, and O. Roustant, Additive covariance kernels for high-dimensional Gaussian process modeling, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 3, 481–499. MR 3076409
- A. M. Dykhne, Conductivity of a two-dimensional two-phase system, Zh. Eksp. Teor. Fiz. 59 (1970), 110–115.
- A. Gandolfi, M. Keane, and L. Russo, On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation, Ann. Probab. 16 (1988), no. 3, 1147–1157. MR 942759
- C. Garban and H. Vanneuville, Bargmann-Fock percolation is noise sensitive, Electron. J. Probab. 25 (2020), Paper No. 98, 20. MR 4136478
- G. Grimmett, Percolation, second ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, Springer-Verlag, Berlin, 1999. MR 1707339
- T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960), 13–20. MR 115221
- M. B. Isichenko, Percolation, statistical topography, and transport in random media, Rev. Modern Phys. 64 (1992), no. 4, 961–1043. MR 1187940
- H. Kesten, The critical probability of bond percolation on the square lattice equals ${1\over 2}$, Comm. Math. Phys. 74 (1980), no. 1, 41–59. MR 575895
- M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics, Springer-Verlag, New York-Berlin, 1983. MR 1004671
- S. A. Molchanov and A. K. Stepanov, Percolation in random fields. I, Teoret. Mat. Fiz. 55 (1983), no. 2, 246–256. MR 734878
- S. A. Molchanov and A. K. Stepanov, Percolation in random fields. II, Teoret. Mat. Fiz. 55 (1983), no. 3, 592–599. MR 734878
- S. Muirhead, A. Rivera, and H. Vanneuville (with an appendix by L. Köhler-Schindler), The phase transition for planar Gaussian percolation models without FKG, arXiv preprint arXiv:2010.11770 (2020).
- S. Muirhead and H. Vanneuville, The sharp phase transition for level set percolation of smooth planar Gaussian fields, Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020), no. 2, 1358–1390. MR 4076787
- G. Pete, Corner percolation on $\mathbb Z^2$ and the square root of 17, Ann. Probab. 36 (2008), no. 5, 1711–1747. MR 2440921
- A. Rivera, Talagrand’s inequality in planar Gaussian field percolation, Electron. J. Probab. 26 (2021), Paper No. 26, 25. MR 4235477
- A. Rivera and H. Vanneuville, The critical threshold for Bargmann–Fock percolation, Ann. H. Lebesgue 3 (2020), 169–215. MR 4060853
- K. Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes, Statist. Probab. Lett. 106 (2015), 239–246. MR 3389997
- R. Zallen and H. Scher, Percolation on a continuum and the localization-delocalization transition in amorphous semiconductors, Phys. Rev. B. 4 (1971), 4471–4479.
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Additional Information
S. Muirhead
Affiliation:
School of Mathematics and Statistics, University of Melbourne
Email:
smui@unimelb.edu.au
Keywords:
Gaussian fields,
level sets,
nodal domains,
percolation
Received by editor(s):
June 24, 2021
Accepted for publication:
October 21, 2021
Published electronically:
May 16, 2022
Additional Notes:
Supported by the Australian Research Council (ARC) Discovery Early Career Researcher Award DE200101467.
Article copyright:
© Copyright 2022
Taras Shevchenko National University of Kyiv