×

Kac-Rice formula for transverse intersections. (English) Zbl 1484.60008

Summary: We prove a generalized Kac-Rice formula that, in a well defined regular setting, computes the expected cardinality of the preimage of a submanifold via a random map by expressing it as the integral of a density. Our proof starts from scratch and although it follows the guidelines of the standard proofs of Kac-Rice formula, it contains some new ideas coming from the point of view of measure theory. Generalizing further, we extend this formula to any other type of counting measure, such as the intersection degree. We discuss in depth the specialization to smooth Gaussian random sections of a vector bundle. Here, the formula computes the expected number of points where the section meets a given submanifold of the total space, it holds under natural non-degeneracy conditions and can be simplified by using appropriate connections. Moreover, we point out a class of submanifolds, that we call sub-Gaussian, for which the formula is locally finite and depends continuously with respect to the covariance of the first jet. In particular, this applies to any notion of singularity of sections that can be defined as the set of points where the jet prolongation meets a given semialgebraic submanifold of the jet space. Various examples of applications and special cases are discussed. In particular, we report a new proof of the Poincaré kinematic formula for homogeneous spaces and we observe how the formula simplifies for isotropic Gaussian fields on the sphere.

MSC:

60D05 Geometric probability and stochastic geometry
60G15 Gaussian processes
57N75 General position and transversality
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
58K05 Critical points of functions and mappings on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adler, RJ; Taylor, JE, Random Fields and Geometry (2007), New York: Springer, New York · Zbl 1149.60003
[2] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems (2000), Clarendon Press: Oxford Science Publications, Clarendon Press · Zbl 0957.49001
[3] Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, Volume 1: Classification of Critical Points, Caustics and Wave Fronts. Modern Birkhäuser Classics. Birkhäuser, Boston (2012) · Zbl 1290.58001
[4] Azais, J-M; Wschebor, M., Level Sets and Extrema of Random Processes and Fields (2009), Hoboken: Wiley, Hoboken · Zbl 1168.60002 · doi:10.1002/9780470434642
[5] Billingsley, Patrick, Convergence of Probability Measures (1999), New York: John Wiley and Sons, Inc., New York · Zbl 0944.60003 · doi:10.1002/9780470316962
[6] Bott, R.; Tu, LW, Differential Forms in Algebraic Topology (1995), New York: Springer, New York · Zbl 0496.55001
[7] Breiding, P.; Keneshlou, H.; Lerario, A., Quantitative singularity theory for random polynomials, Int. Math. Res. Not., 10, rnaa274 (2020) · Zbl 1494.14058 · doi:10.1093/imrn/rnaa274
[8] Chavel, I., Riemannian Geometry: A Modern Introduction. Cambridge Studies in Advanced Mathematics (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1099.53001 · doi:10.1017/CBO9780511616822
[9] Çınlar, E., Probability and Stochastics (2011), New York: Springer, New York · Zbl 1226.60001 · doi:10.1007/978-0-387-87859-1
[10] Dudley, RM, Real Analysis and Probability (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1023.60001 · doi:10.1017/CBO9780511755347
[11] Federer, H., Geometric Measure Theory (1996), Grundlehren der mathematischen Wissenschaften: Springer, Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-3-642-62010-2
[12] Fyodorov, YV; Lerario, A.; Lundberg, E., On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys., 95, 1-20 (2015) · Zbl 1367.60005 · doi:10.1016/j.geomphys.2015.04.006
[13] Gayet, D.; Welschinger, J-Y, Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc., 90, 1, 105-120 (2014) · Zbl 1326.14139 · doi:10.1112/jlms/jdu018
[14] Gayet, D.; Welschinger, J-Y, Expected topology of random real algebraic submanifolds, J. Inst. Math. Jussieu, 14, 4, 673-702 (2015) · Zbl 1326.32040 · doi:10.1017/S1474748014000115
[15] Gayet, D.; Welschinger, J-Y, Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, J. Eur. Math. Soc. (JEMS), 18, 4, 733-772 (2016) · Zbl 1408.14187 · doi:10.4171/JEMS/601
[16] Goresky, M.; MacPherson, R., Stratified Morse Theory (1988), Ergebnisse der Mathematik und ihrer Grenzgebiete: Springer-Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete · Zbl 0639.14012 · doi:10.1007/978-3-642-71714-7
[17] Hirsch, M.W.: Differential Topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag, New York, (1994). Corrected reprint of the 1976 original
[18] Howard, R., The kinematic formula in Riemannian homogeneous spaces, Mem. Am. Math. Soc., 106, 509, vi+69 (1993) · Zbl 0810.53057
[19] Jordan, C., Essai sur la géométrie à \(n\) dimensions, Bull. Soc. Math. Fr., 3, 103-174 (1875) · JFM 07.0457.01 · doi:10.24033/bsmf.90
[20] Kac, M., On the average number of real roots of a random algebraic equation, Bull. Am. Math. Soc., 49, 314-320 (1943) · Zbl 0060.28602 · doi:10.1090/S0002-9904-1943-07912-8
[21] Kostlan, Eric: On the distribution of roots of random polynomials. In From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990), pp. 419-431. Springer, New York (1993) · Zbl 0788.60069
[22] Lerario, A., Random matrices and the average topology of the intersection of two quadrics, Proc. Am. Math. Soc., 143, 8, 3239-3251 (2015) · Zbl 1349.60005 · doi:10.1090/proc/12324
[23] Lerario, A.; Lundberg, E., Statistics on Hilbert’s 16th problem, Int. Math. Res. Not. IMRN, 12, 4293-4321 (2015) · Zbl 1396.14049
[24] Lerario, A.; Lundberg, E., On the geometry of random lemniscates, Proc. Lond. Math. Soc., 113, 5, 649-673 (2016) · Zbl 1384.30007 · doi:10.1112/plms/pdw039
[25] Lerario, A., Stecconi, M.: Maximal and typical topology of real polynomial singularities (2019) · Zbl 1489.14080
[26] Lerario, A., Stecconi, M.: Differential topology of gaussian random fields (2021) · Zbl 1489.14080
[27] Marinucci, D.; Peccati, G., Random Fields on the Sphere: Representation (2011), Cambridge: Cambridge University Press, Cambridge · Zbl 1260.60004 · doi:10.1017/CBO9780511751677
[28] Miao, J.; Ben-Israel, A., On principal angles between subspaces in rn, Linear Algebra Appl., 171, 81-98 (1992) · Zbl 0779.15003 · doi:10.1016/0024-3795(92)90251-5
[29] Milnor, J.W., Stasheff, J.D.: Characteristic classes. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, (1974). Annals of Mathematics Studies, No. 76 · Zbl 0298.57008
[30] Nazarov, F.; Sodin, M., Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, Zh. Mat. Fiz. Anal. Geom., 12, 3, 205-278 (2016) · Zbl 1358.60057 · doi:10.15407/mag12.03.205
[31] Nazarov, F.; Sodin, M., On the number of nodal domains of random spherical harmonics, Am. J. Math., 131, 5, 1337-1357 (2009) · Zbl 1186.60022 · doi:10.1353/ajm.0.0070
[32] Nicolaescu, LI, A stochastic gauss-bonnet-chern formula, Probab. Theory Relat. Fields, 165, 1, 235-265 (2016) · Zbl 1346.35140 · doi:10.1007/s00440-015-0630-z
[33] Park, C., Pranav, P., Chingangbam, P., Van De Weygaert, R., Jones, B., Vegter, G., Kim, I., Hidding, J., Hellwing, W.A.: Betti numbers of Gaussian fields (2013)
[34] Parthasarathy, KR, Probability Measures on Metric Spaces (2005), Academic Press: Ams Chelsea Publishing, Academic Press · Zbl 1188.60001
[35] Rice, SO, Mathematical analysis of random noise, Bell Syst. Tech. J., 23, 3, 282-332 (1944) · Zbl 0063.06485 · doi:10.1002/j.1538-7305.1944.tb00874.x
[36] Sarnak, P.; Wigman, I., Topologies of nodal sets of random band-limited functions, Commun. Pure Appl. Math., 72, 2, 275-342 (2019) · Zbl 1414.58019 · doi:10.1002/cpa.21794
[37] Shub, M., Smale, S.: Complexity of Bezout’s theorem. II. Volumes and probabilities. In Computational algebraic geometry (Nice, 1992), volume 109 of Progr. Math., pp. 267-285. Birkhäuser Boston, Boston (1993) · Zbl 0851.65031
[38] Wigman, I., Fluctuations of the nodal length of random spherical harmonics, Commun. Math. Phys., 298, 3, 787-831 (2010) · Zbl 1213.33019 · doi:10.1007/s00220-010-1078-8
[39] Wigman, Igor, On the expected betti numbers of the nodal set of random fields, Anal. pde, 14, 6, 1797-1816 (2020) · Zbl 1481.60096 · doi:10.2140/apde.2021.14.1797
[40] Zhu, P.; Knyazev, AV, Angles between subspaces and their tangents, J. Numer. Math., 21, 4, 325-340 (2013) · Zbl 1286.65052 · doi:10.1515/jnum-2013-0013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.