Volume 7 (2024)
Mathis, Léo;  Stecconi, Michele
Expectation of a random submanifold: the zonoid section
Annales Henri Lebesgue, Volume 7 (2024), pp. 903-967.

Metadata

DOI10.5802/ahl.214
Keywords Kac–Rice Formula, zonoids, random fields, convex bodies

Abstract

We develop a calculus based on zonoids – a special class of convex bodies – for the expectation of functionals related to a random submanifold Z defined as the zero set of a smooth vector valued random field on a Riemannian manifold. We identify a convenient set of hypotheses on the random field under which we define its zonoid section, an assignment of a zonoid ζ(p) in the exterior algebra of the cotangent space at each point p of the manifold. We prove that the first intrinsic volume of ζ(p) is the Kac–Rice density of the expected volume of Z, while its center computes the expected current of integration over Z. We show that the intersection of random submanifolds corresponds to the wedge product of the zonoid sections and that the preimage corresponds to the pull-back.

Combining this with the recently developed zonoid algebra, it allows to give a multiplication structure to the Kac–Rice formulas, resembling that of the cohomology ring of a manifold. Moreover, it establishes a connection with the theory of convex bodies and valuations, which includes deep results such as the Alexandrov–Fenchel inequality and the Brunn–Minkowski inequality. We export them to this context to prove two analogous new inequalities for random submanifolds. Applying our results in the context of Finsler geometry, we prove some new Crofton formulas for the length of curves and the Holmes–Thompson volumes of submanifolds in a Finsler manifold.


References

[AK18] Akhiezer, Dmitri; Kazarnovskii, Boris Ya. Average number of zeros and mixed symplectic volume of Finsler sets, Geom. Funct. Anal., Volume 28 (2018) no. 6, pp. 1517-1547 | DOI | Zbl

[All72] Allard, William K. On the First Variation of a Varifold, Ann. Math., Volume 95 (1972) no. 3, pp. 417-491 | DOI | Zbl

[Anc20] Ancona, Michele Expected number and distribution of critical points of real Lefschetz pencils, Ann. Inst. Fourier, Volume 70 (2020) no. 3, pp. 1085-1113 | DOI | Numdam | Zbl

[AT07] Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry, Springer Monographs in Mathematics, Springer, 2007 | MR | Zbl

[AV75] Artstein, Zvi; Vitale, Richard A. A Strong Law of Large Numbers for Random Compact Sets, Ann. Probab., Volume 3 (1975) no. 5, pp. 879-882 | DOI | Zbl

[AW09] Azaïs, Jean-Marc; Wschebor, Mario Level sets and extrema of random processes and fields, John Wiley & Sons, 2009 | DOI | MR | Zbl

[BBLM22] Breiding, Paul; Bürgisser, Peter; Lerario, Antonio; Mathis, Léo The zonoid algebra, generalized mixed volumes, and random determinants, Adv. Math., Volume 402 (2022), 108361 | DOI | Zbl

[BCS00] Bao, David; Chern, Shiing-Shen; Shen, Zhongmin An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, 200, Springer, 2000 | DOI | MR | Zbl

[Ber77] Berry, Michael V. Regular and irregular semiclassical wavefunctions, J. Phys. A. Math. Gen., Volume 10 (1977) no. 12, pp. 2083-2091 | DOI | Zbl

[Ber07] Bernig, Andreas Valuations with Crofton formula and Finsler geometry, Adv. Math., Volume 210 (2007) no. 2, pp. 733-753 | DOI | Zbl

[BFS14] Bernig, Andreas; Fu, Joseph H. G.; Solanes, Gil Integral geometry of complex space forms, Geom. Funct. Anal., Volume 24 (2014) no. 2, pp. 403-492 | DOI | Zbl

[Bil95] Billingsley, Patrick Probability and measure, Wiley Series in Probability and mathematical Statistics, John Wiley & Sons, 1995 | Zbl

[Bil99] Billingsley, Patrick Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, 1999 (A Wiley-Interscience Publication) | DOI | MR | Zbl

[BKL18] Breiding, Paul; Kozhasov, Khazhgali; Lerario, Antonio On the geometry of the set of symmetric matrices with repeated eigenvalues, Arnold Math. J., Volume 4 (2018) no. 3, pp. 423-443 | DOI | Zbl

[BL20] Bürgisser, Peter; Lerario, Antonio Probabilistic Schubert calculus, J. Reine Angew. Math., Volume 760 (2020), pp. 1-58 | DOI | Zbl

[BLLP19] Basu, Saugata; Lerario, Antonio; Lundberg, Erik; Peterson, Chris Random fields and the enumerative geometry of lines on real and complex hypersurfaces, Math. Ann., Volume 374 (2019) no. 3, pp. 1773-1810 | DOI | Zbl

[Bog98] Bogachev, Vladimir I. Gaussian Measures, Mathematical Surveys and Monographs, 60035, American Mathematical Society, 1998 | DOI | Zbl

[Bre11] Brezis, Haïm Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011 | DOI | Zbl

[BT82] Bott, Raoul; Tu, Loring W. Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82, Springer, 1982 | DOI | Zbl

[CCJ19] Probabilistic Methods in Geometry, Topology and Spectral Theory (Canzani, Yaiza; Chen, Linan; Jakobson, Dmitry, eds.), Contemporary Mathematics, 739, American Mathematical Society, 2019 | DOI | Zbl

[CH20] Canzani, Yaiza; Hanin, Boris Local Universality for Zeros and Critical Points of Monochromatic Random Waves, Commun. Math. Phys., Volume 378 (2020) no. 3, pp. 1677-1712 | DOI | Zbl

[CM15] Cammarota, Valentina; Marinucci, Domenico On the limiting behaviour of needlets polyspectra, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 3, pp. 1159-1189 | DOI | Numdam | MR | Zbl

[CM18] Cammarota, Valentina; Marinucci, Domenico A quantitative central limit theorem for the Euler-–Poincaré characteristic of random spherical eigenfunctions, Ann. Probab., Volume 46 (2018) no. 6, pp. 3188-3228 | DOI | Zbl

[DMS12] Dinh, Tien-Cuong; Marinescu, George; Schmidt, Viktoria Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting, J. Stat. Phys., Volume 148 (2012) no. 1, pp. 113-136 | DOI | Zbl

[DR18] Dang, Nguyen Viet; Rivière, Gabriel Equidistribution of the conormal cycle of random nodal sets, J. Eur. Math. Soc. (2018) | DOI | Zbl

[Dud02] Dudley, Richard M. Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press, 2002 | DOI | Zbl

[FLL15] Fyodorov, Yan V.; Lerario, Antonio; Lundberg, Erik On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys., Volume 95 (2015), pp. 1-20 | DOI | MR | Zbl

[Gas20] Gass, Louis Almost sure asymptotics for Riemannian random waves (2020) | arXiv

[GW14] Gayet, Damien; Welschinger, Jean-Yves Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc., Volume 90 (2014) no. 1, pp. 105-120 | DOI | MR | Zbl

[GW15] Gayet, Damien; Welschinger, Jean-Yves Expected topology of random real algebraic submanifolds, J. Inst. Math. Jussieu, Volume 14 (2015) no. 4, pp. 673-702 | DOI | MR | Zbl

[GW16] Gayet, Damien; Welschinger, Jean-Yves Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc., Volume 18 (2016) no. 4, pp. 733-772 | DOI | MR | Zbl

[Hau14] Hausdorff, Felix Grundzüge der mengenlehre, Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Series, Veit & Comp, Leipzig, 1914

[Hir76] Hirsch, Morris W. Differential topology, Graduate Texts in Mathematics, 33, Springer, 1976 (corrected reprint of the 1976 original) | DOI | MR | Zbl

[Kaz20] Kazarnovskii, Boris Ya. Average Number of Roots of Systems of Equations, Funct. Anal. Appl., Volume 54 (2020) no. 2, pp. 100-109 | DOI | Zbl

[KKW13] Krishnapur, Manjunath; Kurlberg, Par; Wigman, Igor Nodal length fluctuations for arithmetic random waves, Ann. Math., Volume 177 (2013) no. 2, pp. 699-737 | DOI | Zbl

[KL20] Kozhasov, Khazhgali; Lerario, Antonio On the number of flats tangent to convex hypersurfaces in random position, Discrete Comput. Geom., Volume 63 (2020) no. 1, pp. 229-254 | DOI | Zbl

[Kos93] Kostlan, Eric On the distribution of roots of random polynomials, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), Springer, 1993, pp. 419-431 | DOI | MR | Zbl

[KSW21] Kabluchko, Zakhar; Sartori, Andrea; Wigman, Igor Expected nodal volume for non-Gaussian random band-limited functions (2021) | arXiv

[KWY21] Kurlberg, Par; Wigman, Igor; Yesha, Nadav The defect of toral Laplace eigenfunctions and Arithmetic Random Waves: Toral defect, Nonlinearity (2021) | DOI | Zbl

[Let16] Letendre, Thomas Expected volume and Euler characteristic of random submanifolds, J. Funct. Anal., Volume 270 (2016) no. 8, pp. 3047-3110 | DOI | Zbl

[LL16a] Lerario, Antonio; Lundberg, Erik Gap probabilities and Betti numbers of a random intersection of quadrics, Discrete Comput. Geom., Volume 55 (2016) no. 2, pp. 462-496 | DOI | MR | Zbl

[LL16b] Lerario, Antonio; Lundberg, Erik On the geometry of random lemniscates, Proc. Lond. Math. Soc., Volume 113 (2016) no. 5, pp. 649-673 | DOI | MR | Zbl

[LM21] Lerario, Antonio; Mathis, Leo Probabilistic Schubert Calculus: Asymptotics, Arnold Math. J., Volume 7 (2021) no. 2, pp. 169-194 | DOI | Zbl

[LS19a] Lerario, A.; Stecconi, M. Maximal and Typical Topology of Real Polynomial Singularities (2019) (in press to be published in Annales de l’Institut Fourier) | arXiv

[LS19b] Lerario, Antonio; Stecconi, Michele Differential Topology of Gaussian Random Fields (2019) | arXiv

[Maf17] Maffucci, Riccardo W. Nodal intersections for random waves against a segment on the 3-dimensional torus, J. Funct. Anal., Volume 272 (2017) no. 12, pp. 5218-5254 | DOI | Zbl

[Mar21] Marinucci, Domenico Some Recent Developments on the Geometry of Random Spherical Eigenfunctions (2021) | arXiv

[MP11] Marinucci, Domenico; Peccati, Giovanni Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications, London Mathematical Society Lecture Note Series, Cambridge University Press, 2011 | DOI | Zbl

[MPRW15] Marinucci, Domenico; Peccati, Giovanni; Rossi, Maurizia; Wigman, Igor Non-Universality of Nodal Length Distribution for Arithmetic Random Waves, Geom. Funct. Anal., Volume 26 (2015), pp. 926-960 | DOI | Zbl

[MRV21] Marinucci, Domenico; Rossi, Maurizia; Vidotto, Anna Non-universal fluctuations of the empirical measure for isotropic stationary fields on 𝕊 2 ×ℝ, Ann. Appl. Probab., Volume 31 (2021) no. 5, pp. 2311-2349 | DOI | Zbl

[MRW20] Marinucci, Domenico; Rossi, Maurizia; Wigman, Igor The asymptotic equivalence of the sample trispectrum and the nodal length for random spherical harmonics, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 1, pp. 374-390 | DOI | Zbl

[MSS14] Molchanov, Ilya; Schmutz, Michael; Stucki, Kaspar Invariance properties of random vectors and stochastic processes based on the zonoid concept, Bernoulli, Volume 20 (2014) no. 3, pp. 1210-1233 | DOI | Zbl

[MW11a] Marinucci, Domenico; Wigman, Igor The defect variance of random spherical harmonics, J. Phys. A. Math. Theor., Volume 44 (2011) no. 35, 355206 | DOI | Zbl

[MW11b] Marinucci, Domenico; Wigman, Igor On the area of excursion sets of spherical Gaussian eigenfunctions, J. Math. Phys., Volume 52 (2011) no. 9, 093301 | DOI | Zbl

[MW14] Marinucci, Domenico; Wigman, Igor On Nonlinear Functionals of Random Spherical Eigenfunctions, Commun. Math. Phys., Volume 327 (2014) | DOI | Zbl

[Nic16] Nicolaescu, Liviu I. A stochastic Gauss–Bonnet–Chern formula, Probab. Theory Relat. Fields, Volume 165 (2016) no. 1, pp. 235-265 | DOI | Zbl

[Nic20] Nicolaescu, Liviu I. Lectures On The Geometry Of Manifolds, World Scientific, 2020 | DOI | Zbl

[Not21] Notarnicola, Massimo Matrix Hermite polynomials, Random determinants and the geometry of Gaussian fields (2021) | arXiv

[NPR19] Nourdin, Ivan; Peccati, Giovanni; Rossi, Maurizia Nodal statistics of planar random waves, Commun. Math. Phys., Volume 369 (2019) no. 1, pp. 99-151 | DOI | MR | Zbl

[NS09] Nazarov, Fedor L.; Sodin, Mikhail On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009) no. 5, pp. 1337-1357 | DOI | MR | Zbl

[NS16a] Nazarov, Fedor L.; Sodin, Mikhail Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, J. Math. Phys. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | DOI | MR | Zbl

[NS16b] Nicolaescu, Liviu I.; Savale, Nikhil The Gauss–Bonnet–Chern theorem: a probabilistic perspective, Probab. Theory Relat. Fields, Volume 369 (2016) no. 4, pp. 2951-2986 | DOI | Zbl

[Par05] Parthasarathy, Kalyanapuram R. Probability Measures on Metric Spaces, AMS Chelsea Publishing; Academic Press Inc., 2005 | Zbl

[PF08] Paiva, Juan C. Á.; Fernandes, Emmanuel Gelfand transforms and Crofton formulas, Sel. Math., New Ser., Volume 13 (2008) no. 3, p. 369 | DOI | Zbl

[RW16] Rudnick, Zeev; Wigman, Igor Nodal intersections for random eigenfunctions on the torus, Am. J. Math., Volume 138 (2016) no. 6, pp. 1605-1644 | DOI | Zbl

[Sar42] Sard, Arthur The measure of the critical values of differentiable maps, Bull. Am. Math. Soc., Volume 48 (1942) no. 12, pp. 883-890 | DOI | Zbl

[Sch01] Schneider, Rolf Crofton formulas in hypermetric projective Finsler spaces, Arch. Math., Volume 77 (2001) no. 1, pp. 85-97 | DOI | Zbl

[Sch14] Schneider, Rolf Convex bodies: the Brunn–Minkowski theory, Encyclopedia of Mathematics and Its Applications, 151, Cambridge University Press, 2014 | MR | Zbl

[Spi79] Spivak, Michael A comprehensive introduction to differential geometry. Vol. I, Publish or Perish, Inc., Wilmington, Del., 1979 | MR | Zbl

[SS93a] Shub, Michael; Smale, Stephen Complexity of Bezout’s theorem. II. Volumes and probabilities, Computational algebraic geometry (Nice, 1992) (Progress in Mathematics), Volume 109, Birkhäuser, 1993, pp. 267-285 | DOI | MR | Zbl

[SS93b] Shub, Michael; Smale, Steve Complexity of Bézout’s theorem. I. Geometric aspects, J. Am. Math. Soc., Volume 6 (1993) no. 2, pp. 459-501 | DOI | MR | Zbl

[SS93c] Shub, Michael; Smale, Steve Complexity of Bezout’s theorem. III. Condition number and packing, J. Complexity, Volume 9 (1993) no. 1, pp. 4-14 (Festschrift for Joseph F. Traub, Part I) | DOI | MR | Zbl

[Ste21] Stecconi, Michele Isotropic Random Spin Weighted Functions on 𝕊 2 vs Isotropic Random Fields on 𝕊 3 (2021) (in press, to be published in Theory of Probability and Mathematical Statistics) | arXiv

[Ste22] Stecconi, Michele Kac–Rice formula for transverse intersections, Anal. Math. Phys., Volume 12 (2022) no. 2, 44 | DOI | Zbl

[SW19] Sarnak, Peter; Wigman, Igor Topologies of nodal sets of random band-limited functions, Commun. Pure Appl. Math., Volume 72 (2019) no. 2, pp. 275-342 | DOI | MR | Zbl

[SZ99] Shiffman, Bernard; Zelditch, Steve Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys., Volume 200 (1999) no. 3, pp. 661-683 | DOI | Zbl

[SZ08] Shiffman, Bernard; Zelditch, Steve Number Variance of Random Zeros on Complex Manifolds, Geom. Funct. Anal., Volume 18 (2008) no. 4, pp. 1422-1475 | DOI | Zbl

[Vit91] Vitale, Richard A. Expected absolute random determinants and zonoids, Ann. Appl. Probab., Volume 1 (1991) no. 2, pp. 293-300 | MR | Zbl

[Whi35] Whitney, Hassler A function not constant on a connected set of critical points, Duke Math. J., Volume 1 (1935) no. 4, pp. 514-517 | DOI | Zbl

[Wig10] Wigman, Igor Fluctuations of the Nodal Length of Random Spherical Harmonics, Commun. Math. Phys., Volume 298 (2010) no. 3, pp. 787-831 | DOI | Zbl

[Wig11] Wigman, Igor On the nodal lines of random and deterministic Laplace eigenfunctions (2011) | arXiv

[Wig22] Wigman, Igor On the nodal structures of random fields – a decade of results (2022) | arXiv

[Zel09] Zelditch, Steve Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory (Contemporary Mathematics), Volume 484, American Mathematical Society, 2009, pp. 321-342 | DOI | Zbl

[ÀPB10] Àlvarez-Paiva, Juan-Carlos; Berck, Gautier Finsler surfaces with prescribed geodesics (2010) | arXiv

[ÀPT04] Àlvarez-Paiva, Juan-Carlos; Thompson, Anthony C. Volumes on normed and Finsler spaces, A sampler of Riemann–Finsler geometry (Mathematical Sciences Research Institute Publications), Volume 50, Cambridge University Press, 2004, pp. 1-48 | DOI | MR | Zbl

[Çın11] Çınlar, E. Probability and Stochastics, Graduate Texts in Mathematics, 261, Springer, 2011 | DOI | Zbl