2005/08/31 by Béla Bollobás, Bela Bollobas, Svante Janson +1 · 1 citation
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:05C80 #msc:60K35
paper · pdf · doi:10.1002/rsa.20175
published as Random Struct. Algorithms 31 (2007), 239-246. · 9 pages. Title changed. Minor changes to text, including updated references to [3]. To appear in Random Structures and Algorithms
arxiv created 2006/10/16 · openalex publication_date 2007/04/09 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Abstract Fix d ≥ 2, and let X be either ℤ d or the points of a Poisson process in ℝ d of intensity 1. Given parameters r and p , join each pair of points of X within distance r independently with probability p . This is the simplest case of a “spread‐out” percolation model studied by Penrose [Ann Appl Probab 3 (1993) 253–276], who showed that, as r → ∞ , the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of [3] on inhomogeneous random graphs. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007