In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of ffGW + ffl, for all ffl> 0; here ffGW ss.878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [25]. This implies that if the Unique Games

Abstract. In their seminal work which initiated random graph theory Erdös and Rényi discovered that many graph properties have sharp thresholds as the number of vertices tends to infinity. We prove a conjecture of Linial that every monotone graph property has a sharp threshold. This follows from the following theorem. Let Vn(p) ={0,1} n denote the Hamming space endowed with the probability measure µp defined by µp(ɛ1,ɛ2,...,ɛn) = pk ·(1 − p) n−k,where k = ɛ1+ ɛ2+ ···+ ɛn. Let A be a monotone subset of Vn. We say that A is symmetric if there is a transitive permutation group Γ on {1, 2,...,n} such that A is invariant under Γ. Theorem. For every symmetric monotone A,ifµp(A)>ɛthen µq(A)> 1−ɛ for q = p + c1 log(1/2ɛ) / log n. (c1isan absolute constant.) 1. Graph properties A graph property is a property of graphs which depends only on their isomorphism class. Let P be a monotone graph property; that is, if a graph G satisfies P

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an n + 1 by n grid. A configuration is a function that assigns to every edge the value 0 or 1. Let ω be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges e with ω(e) = 1. By duality, the probability for having a crossing is 1/2. Fix an ǫ ∈ (0,1). For each edge e, let ω ′ (e) = ω(e) with probability 1 − ǫ, and ω ′ (e) = 1 − ω(e)

We present a (randomized) test for monotonicity of Boolean functions. Namely, given the ability to query an unknown function f : f0; 1g 7! f0; 1g at arguments of its choice, the test always accepts a monotone f , and rejects f with high probability if it is ffl-far from being monotone (i.e., every monotone function differs from f on more than an ffl fraction of the domain).

We introduce a natural k-coloring algorithm and analyze its performance on random graphs with con-stant expected degree c (Gn,p=c/n). For k = 3 our re-sults imply that almost all graphs with n vertices and 1.923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability significantly and settles the last case of Q long-standing open question of Bollobas [5]. We also provide a tight asymptotic analysis of the algorithm. We show that for all k 2 3, if c 5 klnk- 3/2k then the algorithm almost surely succeeds, while for any E> 0, and k sufficiently large, if c 2 (1 + E)k In k then the algorithm almost surely fails. The analysis is based on the use of differential equations to approximate the mean path of certain Markov chains.

Introduction. A subset A of f0; 1g n is called monotone provided if x 2 A; x 0 2 f0; 1g n ; x i x 0 i for i = 1; : : : ; n then x 0 2 A. For 0 p 1, define p the product measure on f0; 1g n with weights 1 \Gamma p at 0 and p at 1. Thus p (fxg) = (1 \Gamma p) n\Gammaj p j where j = #fi = 1; : : : ; njx i = 1g: (0.1) If

We study percolation in the following random environment: let Z be a Poisson process of constant intensity on R 2, and form the Voronoi tessellation of R 2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p> 1/2 then the union of the black cells contains an infinite component with probability 1, while if p < 1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten’s results for bond percolation in Z 2. The result corresponding to Harris ’ Theorem for bond percolation in Z 2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten’s results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten’s Theorem for Z 2; we hope

Abstract- The dimme drctrrbutron of a binary code C ' is the sequence

Let 0 < a < b < ∞, and for each edge e of Z d let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d> 1 the distance distω(0,v) from the origin to a vertex v, |v |> 2, has variance bounded by C |v|/log |v|, where C = C(a,b,d) is a constant which may only depend on a, b and d. Some related variants are also discussed. 1

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