×

Poisson and Gaussian fluctuations for the components of the \(\mathbf{f}\)-vector of high-dimensional random simplicial complexes. (English) Zbl 1453.60014

Summary: We investigate the high-dimensional asymptotic distributional behavior of the components of the \(\mathbf{f}\)-vector of a random Vietoris-Rips complex that is generated over a Poisson point process in \([-\frac{1}{2},\frac{1}{2}]^d\) as the space dimension and the intensity tend to infinity while the radius parameter tends to zero simultaneously.

MSC:

60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
55U10 Simplicial sets and complexes in algebraic topology
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] Bobrowski, O. and Kahle, M. Topology of random geometric complexes: a survey. J. Appl. Comput. Topol.,1(3-4), 331-364 (2018) · Zbl 1402.60015
[2] Bourguin, S. and Peccati, G. The Malliavin-Stein method on the Poisson space. InStochastic analysis for Poisson point processes, volume 7 ofBocconi Springer Ser., pp. 185-228. Bocconi Univ. Press (2016) · Zbl 1350.60005
[3] Grygierek, J. Poisson fluctuations for edge counts in high-dimensional random geometric graphs.ArXiv Mathematics e-prints(2019) · Zbl 1453.60015
[4] Grygierek, J. and Thäle, C.Gaussian fluctuations for edge counts in highdimensional random geometric graphs.Statist. Probab. Lett.,158, 108674, 10 (2020) · Zbl 1453.60015
[5] Last, G., Peccati, G., and Schulte, M. Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization.Probab. Theory Related Fields,165(3-4), 667-723 (2016) · Zbl 1347.60012
[6] Last, G. and Penrose, M.Lectures on the Poisson process, volume 7 ofInstitute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge (2018). ISBN 978-1-107-45843-7; 978-1-107-08801-6 · Zbl 1392.60004
[7] Li, S. Concise formulas for the area and volume of a hyperspherical cap.Asian J. Math. Stat.,4(1), 66-70 (2011)
[8] Munkres, J. R.Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA (1984). ISBN 0-201-04586-9 · Zbl 0673.55001
[9] Peccati, G. and Reitzner, M., editors.Stochastic analysis for Poisson point processes, volume 7 ofBocconi & Springer Series. Bocconi University Press (2016) · Zbl 1350.60005
[10] Penrose, M.Random geometric graphs, volume 5 ofOxford Studies in Probability. Oxford University Press, Oxford (2003). ISBN 0-19-850626-0 · Zbl 1029.60007
[11] Reitzner, M. and Schulte, M. Central limit theorems forU-statistics of Poisson point processes.Ann. Probab.,41(6), 3879-3909 (2013) · Zbl 1293.60061
[12] Stanley, R. P.Combinatorics and commutative algebra, volume 41 ofProgress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, second edition (1996). ISBN 0-8176-3836-9 · Zbl 0838.13008
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.