Sharp Thresholds
نویسنده
چکیده
The Kolmogorov Zero-One law states that for tail events on infinite-dimensional probability spaces, the probability must be either zero or one. Behavior analogous in a natural sense to this exists on finite-dimensional spaces as well. Events exhibiting this behavior are said to have a sharp threshold. Let Qn = {0, 1}n be the discrete hypercube with the probability measure Pp defined by Pp(ω1, ω2, . . . , ωn) = pk(1−p)n−k where k = ω1 +ω2 + · · ·+ωn . An up event A ⊂ Qn is symmetric if there exists a transitive permutation group on the indices under which A is invariant. We present the following result, due to Friedgut and Kalai: Theorem 0.1. For every symmetric up-event A ⊂ Qn, if Pp(A) > , then Pq(A) > 1− for q = p+ c log(1/2 )/ log n, where c is an absolute constant (it does not depend on any of the other terms).
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