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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 $f(eta_p)$ flips 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.
A wide array of random graph models have been postulated to understand properties of observed networks. Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges. It is conjectured that for a large class of
Let $mathcal{H}$ denote a collection of subsets of ${1,2,ldots,n}$, and assign independent random variables uniformly distributed over $[0,1]$ to the $n$ elements. Declare an element $p$-present if its corresponding value is at most $p$. In this pape
Scaling limits are analyzed for stochastic continuous opinion dynamics systems, also known as gossip models. In such models, agents update their vector-valued opinion to a convex combination (possibly agent- and opinion-dependent) of their current va
Fix a space dimension $dge 2$, parameters $alpha > -1$ and $beta ge 1$, and let $gamma_{d,alpha, beta}$ be the probability measure of an isotropic random vector in $mathbb{R}^d$ with density proportional to begin{align*} ||x||^alpha, expleft(-frac{|x
The aim of this paper is to give a precise estimate on the tail probability of the visibility function in a germ-grain model: this function is defined as the length of the longest ray starting at the origin that does not intersect an obstacle in a Bo