Abstract
We establish the strong law of large numbers for Betti numbers of random Čech complexes built on \({\mathbb {R}}^N\)-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on \({\mathbb {R}}^N\) and the case where it is supported on a \(C^1\) compact manifold of dimension strictly less than N. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to \(L^p\) spaces, for all \(1\le p < \infty \).
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Acknowledgements
The authors are thankful to Prof. Tomoyuki Shirai for many useful discussions. This work is partially supported by JST CREST Mathematics (15656429). A.G. is fully supported by JICA-Friendship Scholarship. K.D.T. is partially supported by JSPS KAKENHI Grant Numbers JP16K17616. K.T. is partially supported by JSPS KAKENHI Grant Numbers 18K13426.
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Goel, A., Trinh, K.D. & Tsunoda, K. Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime. J Stat Phys 174, 865–892 (2019). https://doi.org/10.1007/s10955-018-2201-z
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DOI: https://doi.org/10.1007/s10955-018-2201-z