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• W3109566025 abstract "Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1le k le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing." @default.
• W3109566025 created "2020-12-07" @default.
• W3109566025 creator A5049296186 @default.
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• W3109566025 date "2020-12-04" @default.
• W3109566025 modified "2023-10-10" @default.
• W3109566025 title "Homological Percolation: The Formation of Giant <i>k</i>-Cycles" @default.
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• W3109566025 doi "https://doi.org/10.1093/imrn/rnaa305" @default.
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