Journal of the Mathematical Society of Japan

Asymptotic behavior of lifetime sums for random simplicial complex processes

Masanori HINO and Shu KANAZAWA

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Abstract

We study the homological properties of random simplicial complexes. In particular, we obtain the asymptotic behavior of lifetime sums for a class of increasing random simplicial complexes; this result is a higher-dimensional counterpart of Frieze's $\zeta(3)$-limit theorem for the Erdős–Rényi graph process. The main results include solutions to questions posed in an earlier study by Hiraoka and Shirai about the Linial–Meshulam complex process and the random clique complex process. One of the key elements of the arguments is a new upper bound on the Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem.

Note

This study was supported by JSPS KAKENHI Grant Number JP15H03625.

Article information

Source
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 765-804.

Dates
Received: 5 February 2018
First available in Project Euclid: 24 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1556092819

Digital Object Identifier
doi:10.2969/jmsj/79777977

Mathematical Reviews number (MathSciNet)
MR3984242

Zentralblatt MATH identifier
07121553

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 55U10: Simplicial sets and complexes 05E45: Combinatorial aspects of simplicial complexes 60C05: Combinatorial probability

Keywords
Linial–Meshulam complex process random clique complex process multi-parameter random simplicial complex lifetime sum Betti number

Citation

HINO, Masanori; KANAZAWA, Shu. Asymptotic behavior of lifetime sums for random simplicial complex processes. J. Math. Soc. Japan 71 (2019), no. 3, 765--804. doi:10.2969/jmsj/79777977. https://projecteuclid.org/euclid.jmsj/1556092819


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References

  • [1] W. Ballmann and J. Świątkowski, On $L^2$-cohomology and property (T) for automorphism groups of polyhedral cell complexes, Geom. Funct. Anal., 7 (1997), 615–645.
  • [2] A. Costa and M. Farber, Random simplicial complexes, In: Configuration spaces, Springer INdAM Ser., 14, Springer, 2016, 129–153.
  • [3] H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, Discrete Comput. Geom., 28 (2002), 511–533.
  • [4] P. Erdős and A. Rényi, On random graphs, Publ. Math. Debrecen, 6 (1959), 290–297.
  • [5] P. Erdős and A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungarian Acad. Sci., 5A (1960), 17–61.
  • [6] C. F. Fowler, Generalized random simplicial complexes, arXiv:1503.01831.
  • [7] A. M. Frieze, On the value of a random minimum spanning tree problem, Discrete Applied Math., 10 (1985), 47–56.
  • [8] H. Garland, $p$-adic curvature and the cohomology of discrete subgroups of $p$-adic groups, Ann. of Math. (2), 97 (1973), 375–423.
  • [9] E. N. Gilbert, Random graphs, Ann. Math. Statist., 30 (1959), 1141–1144.
  • [10] Y. Hiraoka and T. Shirai, Minimum spanning acycle and lifetime of persistent homology in the Linial–Meshulam process, Random Structures Algorithms, 51 (2017), 315–340.
  • [11] C. Hoffman, M. Kahle and E. Paquette, Spectral gaps of random graphs and applications, arXiv:1201.0425.
  • [12] M. Kahle, Topology of random clique complexes, Discrete Math., 309 (2009), 1658–1671.
  • [13] M. Kahle, Sharp vanishing thresholds for cohomology of random flag complexes, Ann. of Math. (2), 179 (2014), 1085–1107.
  • [14] N. Linial and R. Meshulam, Homological connectivity of random 2-complexes, Combinatorica, 26 (2006), 475–487.
  • [15] N. Linial and Y. Peled, On the phase transition in random simplicial complexes, Ann. of Math. (2), 184 (2016), 745–773.
  • [16] R. Meshulam and N. Wallach, Homological connectivity of random $k$-dimensional complexes, Random Structures Algorithms, 34 (2009), 408–417.
  • [17] A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete Comput. Geom., 33 (2005), 249–274.