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On near‐critical and dynamical percolation in the tree case
Author(s) -
Häggström Olle,
Pemantle Robin
Publication year - 1999
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/(sici)1098-2418(199910/12)15:3/4<311::aid-rsa7>3.0.co;2-6
Subject(s) - percolation (cognitive psychology) , tree (set theory) , root (linguistics) , continuum percolation theory , statistical physics , mathematics , combinatorics , cluster (spacecraft) , percolation critical exponents , physics , critical exponent , discrete mathematics , computer science , phase transition , quantum mechanics , linguistics , philosophy , neuroscience , programming language , biology
Consider independent bond percolation with retention probability p on a spherically symmetric tree Γ. Write θ Γ ( p ) for the probability that the root is in an infinite open cluster, and define the critical value p c =inf{ p : θ Γ ( p )>0}. If θ Γ ( p c )=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p c , as demonstrated recently by Häggström, Peres, and Steif. Here we relate this phenomenon to the near‐critical behavior of θ Γ ( p ) by showing that the root percolates in the dynamical percolation process if and only if ∫ p c1 (θ Γ ( p )) −1 dp <∞. The “only if” direction extends to general trees, whereas the “if” direction fails in this generality. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 15, 311–318, 1999