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Testing a property P of graphs in the bounded degree model deals with the following problem: given a graph G of bounded degree d we should distinguish (with probability 0.9, say) between the case that G satisfies P and the case that one should add/remove at least ε d n edges of G to make it satisfy P. In sharp contrast to property testing of dense(More)
We prove that for any decision tree calculating a boolean function f : {−1, 1} n → {−1, 1}, Var[f ] ≤ n 𼇐 i=1 δ i Inf i (f), where δ i is the probability that the ith input variable is read and Inf i (f) is the influence of the ith variable on f. The variance, influence and probability are taken with respect to an arbitrary product measure on {−1, 1} n. It(More)
A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Z t } (with Z 0 chosen according to the stationary distribution) and any map f from the state space to X, the distance D t from f (Z 0) to f (Z t) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for(More)
The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ − 6, and vice versa. We also write down the martingales describing the Radon–Nykodim(More)
The game of Hex has two players who take turns placing stones of their colors on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a crossing between two opposite edges, while White wins by completing a crossing between the other pair of opposite edges. Although ordinary Hex is famously difficult to analyze, random-turn Hex—in which(More)
A Boolean function of <i>n</i> bits is balanced if it takes the value 1 with probability 1&#8260;2. We exhibit a balanced Boolean function with a randomized evaluation procedure (with probability 0 of making a mistake) so that on uniformly random inputs, no input bit is read with probability more than &#920;(<i>n</i><sup>-1/2</sup>&#8730; log <i>n</i>). We(More)
The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract Consider an instance h of the Gaussian free field on a simply connected planar domain D with boundary conditions −λ on one boundary arc and λ on the complementary arc, where λ is the special constant π/8. We argue that even though(More)
Let ; act on a countable set V with only nitely many orbits. Given a ;-invariant random environment for a Markov chain on V and a random scenery, w e exhibit, under certain conditions, an equivalent stationary measure for the environment and scenery from the viewpoint of the random walker. Such theorems have been very useful in investigations of percolation(More)