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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

Basic Technique Disjoint Path Reliability Theory Open Path Open Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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