2020/05/28 by Omer Bobrowski, Bobrowski, Omer, Primož Škraba +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #55U10 #60D05 #60K35 #82B43 #Algebraic Topology (math.AT) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2005.14011
openalex publication_date 2020/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (k=1,...,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.