Volatility of Boolean functions

Journal article, 2016

We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2,..., let f(n) : {0, 1}(mn) -> {0, 1} be a Boolean function and X-(n)(t) = (Xi (t),..., X-mn (t))(t) (is an element of) ([0, infinity)) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate p(n) is an element of [0, 1] and from 1 to 0 with rate q(n) = 1 p(n). Our object of study will be Cn which is the number of state changes of f(n)(X-(n)(t)) as a function oft during [0, 1]. We say that the family {f(n)}(n >= 1) is volatile if Cn -> infinity in distribution as n -> infinity and say that {f(n)}(n >= 1) is tame if {Cn}(n >= 1) is tight. We study these concepts in and of themselves as well as investigate their relationship with the recent notions of noise sensitivity and noise stability. In addition, we study the question of lameness which means that P(C-n = 0) -> 1 as n -> infinity . Finally, we investigate these properties for the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees in various regimes. (C) 2016 Published by Elsevier B.V.