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Universality in two‐dimensional enhancement percolation
Author(s) -
Camia Federico
Publication year - 2008
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/rsa.20220
Subject(s) - percolation critical exponents , percolation threshold , universality (dynamical systems) , monotonic function , directed percolation , continuum percolation theory , bernoulli's principle , statistical physics , scaling , mathematics , critical exponent , scaling limit , critical point (mathematics) , physics , condensed matter physics , mathematical analysis , quantum mechanics , thermodynamics , geometry , electrical resistivity and conductivity
Abstract We consider a type of dependent percolation introduced in 2, where it is shown that certain “enhancements” of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this study we first prove that, for two‐dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two‐dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit 12, 13 is not affected by any monotonic enhancement that does not shift the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008