Percolation pp 32-52 | Cite as

Some Basic Techniques

  • Geoffrey Grimmett
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 321)

Abstract

The probability θ(p), that the origin belongs to an infinite open cluster, is a non-decreasing function of p. This is intuitively obvious since an increase in the value of p leads generally to an increase in the number of open edges of L d , thereby increasing the number and lengths of open paths from the origin. Another way of putting this is to say that {|C| = ∞} is an increasing event, in the sense that: if ω ∈ {|C|=∞} then ω′ ∈ {|C|=∞} whenever ωω′ With such an example in mind we make the following definition. The event A in ℱ is called increasing if I A (ω) ≤ I A (ω′) whenever ωω′, where I A is the indicator function of A. We call A decreasing if its complement A is increasing.

Keywords

Covariance Hunt 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Geoffrey Grimmett
    • 1
  1. 1.Statistical LaboratoryUniversity of CambridgeCambridgeUK

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