Arbitrary Threshold Widths for Monotone, Symmetric Properties

Abstract

We show that there exist symmetric properties in the discrete n-cube whose threshold widths range asymptotically between 1/ √ n and 1/log n. These properties are built using a combination of failure sets arising in reliability theory. This combination of sets is simply called a product. Some general results on the threshold width of the product of two sets A and B in terms of the threshold locations and widths of A and B are provided.

Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome?

“…Moreover, the centralizing coefficient of Theorem 3 could in greater generality be replaced by any sequence (b n ) n≥1 bounded away from 0 and 1, although this may be of less interest. We further mention that Rossignol [45] has previously showed that for any sufficiently smooth sequence (a n ) n≥1 satisfying log n ≤ a n ≤ √ n there exists an increasing sequence (N (n)) n≥1 and monotone and transitive Boolean functions (f N (n) ) n≥1 with a threshold at 1/2 of width 1/a n .…”