Volatility of Boolean functions

We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2, . . ., let fn : {0, 1}mn → {0, 1} be a Boolean function and X(n)(t) = (X1(t), . . . , Xmn(t))t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate pn ∈ [0, 1] and from 1 to 0 with rate qn = 1−pn. Our object of study will be Cn which is the number of state changes of fn(X (n)(t)) as a function of t during [0, 1]. We say that the family {fn}n≥1 is volatile if Cn → ∞ in distribution as n → ∞ and say that {fn}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(Cn = 0)→ 1 as n→∞. Finally, we investigate these properties for a number of standard Boolean functions such as the majority function, iterated 3-majority, tribes, connectivity of the G(n, p) random graph and percolation on certain trees at various levels of the parameter pn. AMS Subject classification : 60K99