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.