2008/06/30 by Michael Damron, Artëm Sapozhnikov, Bálint Vágvölgyi · 1 citation
Mathematics · Physics and Astronomy · #Cluster (spacecraft) #Combinatorics #Complex Network Analysis Techniques #Continuum percolation theory #Critical exponent #Critical point (mathematics) #Disjoint sets #Geometry #Mathematical analysis #Mathematics #Measure (data warehouse) #Percolation (cognitive psychology) #Percolation critical exponents #Percolation threshold #Physics #Quantum mechanics #RADIUS #Statistical physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60K35 #msc:82B43
paper · pdf · doi:10.1214/09-aop462
published as Annals of Probability 2009, Vol. 37, No. 6, 2297-2331 · Published in at http://dx.doi.org/10.1214/09-AOP462 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2009/11/01 · arxiv created 2009/12/09 · arxiv updated 2010/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k≥1, the k-point function of the first so-called pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint pc-open clusters. Further, for k>1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.