We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For $n = 1,2,ldots$, let $f_n:{0,1}^{m_n} ra {0,1}$ be a Boolean function and $X^{(n)}(t)=(X_1(t),ldots,X_{m_n}(t))_{t in [0,infty)}$ be a vec
tor of i.i.d. stationary continuous time Markov chains on ${0,1}$ that jump from $0$ to $1$ with rate $p_n in [0,1]$ and from $1$ to $0$ with rate $q_n=1-p_n$. Our object of study will be $C_n$ which is the number of state changes of $f_n(X^{(n)}(t))$ as a function of $t$ during $[0,1]$. We say that the family ${f_n}_{nge 1}$ is volatile if $C_n ra iy$ in distribution as $ntoinfty$ and say that ${f_n}_{nge 1}$ is tame if ${C_n}_{nge 1}$ 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 $Pro(C_n =0)ra 1$ as $ntoinfty$. Finally, we investigate these properties for a number of standard Boolean functions such as the majority function, iterated 3-majority, the AND/OR function on the binary tree and percolation on certain trees at various levels of the parameter $p_n$.
Given a continuous time Markov Chain ${q(x,y)}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on ${0,1}^S$, which evolves in the following way: (1) for each two sites $x$ and $y$ in $S$, the state at site $
x$ changes to the value of the state at site $y$ at rate $q(x,y)$; (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates ${q(x,y)}$ and the corresponding stationary distributions are almost uniform, then the mixing time has a sharp cutoff at time $log|S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids; we obtain the special case of their result for the cycle as a consequence of our result. Finally, we consider the model on a star and demonstrate the surprising phenomenon that the time it takes for the chain started at all ones to become close in total variation to the chain started at all zeros is of smaller order than the mixing time.
Consider a monotone Boolean function $f:{0,1}^nto{0,1}$ and the canonical monotone coupling ${eta_p:pin[0,1]}$ of an element in ${0,1}^n$ chosen according to product measure with intensity $pin[0,1]$. The random point $pin[0,1]$ where $f(eta_p)$ flip
s from $0$ to $1$ is often concentrated near a particular point, thus exhibiting a threshold phenomenon. For a sequence of such Boolean functions, we peer closely into this threshold window and consider, for large $n$, the limiting distribution (properly normalized to be nondegenerate) of this random point where the Boolean function switches from being 0 to 1. We determine this distribution for a number of the Boolean functions which are typically studied and pay particular attention to the functions corresponding to iterated majority and percolation crossings. It turns out that these limiting distributions have quite varying behavior. In fact, we show that any nondegenerate probability measure on $mathbb{R}$ arises in this way for some sequence of Boolean functions.
