Abstract
We revisit the work of Bourgain et al. (1992) - referred to as "BKKKL" in the title - about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\{1,...,r\}^N$, generalizing one of the main results of Talagrand (1994).
Highlights
The theory of threshold phenomena can be traced back to [Russo, 1982], who described it as an “approximate zero-one law”(see [Margulis, 1974], [Kahn et al, 1988] and [Talagrand, 1994])
We say that an event A ⊂ {0, 1}n is increasing if the indicator function of A is coordinate-wise nondecreasing
The smaller the maximal influence of a coordinate on A is, the smaller is the bound obtained on the length of the interval of values of p
Summary
The theory of threshold phenomena can be traced back to [Russo, 1982], who described it as an “approximate zero-one law”(see [Margulis, 1974], [Kahn et al, 1988] and [Talagrand, 1994]). We say that an event A ⊂ {0, 1}n is increasing if the indicator function of A is coordinate-wise nondecreasing. When the influence of each coordinate on an increasing event A is small (see the definition of γ hereafter), and when the parameter p goes from 0 to one, the probability that A occurs, μp(A), grows from near zero to near one on a short interval of values of p: this is the threshold phenomenon.
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