Percolation pp 32-52 | Cite as

# Some Basic Techniques

## 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 *ω* ≤ *ω*′ 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 *ω* ≤ *ω*′, 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## Preview

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