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Noise sensitivity of Boolean functions and applications to percolation
, 2008
"... 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 ..."
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Cited by 110 (20 self)
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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 noisestable. 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)
Conformally invariant scaling limits: an overview and collection of open problems
 Proceedings of the International Congress of Mathematicians, Madrid (M. SanzSolé et
, 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a oneparameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
Scaling limit of FourierWalsh coefficients (a framework)”, math.PR/9903121
"... Independent random signs can govern various discrete models that converge to nonisomorphic continuous limits. Convergence of FourierWalsh spectra is established under appropriate conditions. ..."
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Independent random signs can govern various discrete models that converge to nonisomorphic continuous limits. Convergence of FourierWalsh spectra is established under appropriate conditions.
Splitting: Tanaka’s SDE revisited
, 1999
"... What follows is my attempt to understand a set of ideas being developed by Boris Tsirelson. I do this by studying a specific, and I hope interesting, example. Tanaka’s SDE is one of the easiest examples of a stochastic differential equation with no strong solution. Suppose ( Xt; t ≥ 0) is a realval ..."
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What follows is my attempt to understand a set of ideas being developed by Boris Tsirelson. I do this by studying a specific, and I hope interesting, example. Tanaka’s SDE is one of the easiest examples of a stochastic differential equation with no strong solution. Suppose ( Xt; t ≥ 0) is a realvalued Brownian motion starting from zero and we put Bt = ∫ t 0 sgn(Xs)dXs then B is also a Brownian motion and Tanaka’s SDE (1) ∫ t
NOISES IN STOCHASTIC PROCESSES
"... 1 Noise stability and sensitivity for stochastic processes represented by noises in discrete time ..."
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1 Noise stability and sensitivity for stochastic processes represented by noises in discrete time