Universality in two-dimensional enhancement percolation

Research output: Contribution to journalArticle

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 monotonie 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.

Original languageEnglish (US)
Pages (from-to)377-408
Number of pages32
JournalRandom Structures and Algorithms
Volume33
Issue number3
DOIs
StatePublished - Oct 1 2008

Keywords

  • Critical exponents
  • Enhancement percolation
  • Scaling limit
  • Universality

ASJC Scopus subject areas

  • Software
  • Mathematics(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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