Sharp thresholds and the partition function

Sharp Thresholds

The Kolmogorov Zero-One law states that for tail events on infinite-dimensional probability spaces, the probability must be either zero or one. Behavior analogous in a natural sense to this exists on finite-dimensional spaces as well. Events exhibiting this behavior are said to have a sharp threshold. Let Qn = {0, 1}n be the discrete hypercube with the probability measure Pp defined by Pp(ω1, ω...

Sharp thresholds and percolation in the plane

Recently, it was shown in [4] that the critical probability for random Voronoi percolation in the plane is 1/2. As a by-product of the method, a short proof of the Harris-Kesten Theorem was given in [5]. The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results, and to prove new ...

Sharp Thresholds of Graph properties and the k sat Problem

Given a monotone graph property P consider p P the proba bility that a random graph with edge probability p will have P The function d p P dp is the key to understanding the threshold behavior of the property P We show that if d p P dp is small corresponding to a non sharp threshold then there is a list of graphs of bounded size such that P can be approximated by the property of having one of t...

On Sharp Thresholds of Monotone Properties

Sharp Load Thresholds for Cuckoo Hashing

The paradigm of many choices has influenced significantly the design of efficient data structures and, most notably, hash tables. Cuckoo hashing is a technique that extends this concept. There, we are given a table with n locations, and we assume that each location can hold one item. Each item to be inserted chooses randomly k ≥ 2 locations and has to be placed in any one of them. How much load...