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Noise sensitivity of Boolean functions and percolation. (English) Zbl 1355.06001
Institute of Mathematical Statistics Textbooks 5. Cambridge: Cambridge University Press (ISBN 978-1-107-07643-3/hbk; 978-1-107-43255-0/pbk; 978-1-139-92416-0/ebook). xvii, 203 p. (2015).
“This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science. Certain functions are highly sensitive to noise; this can be seen via Fourier analysis on the hypercube. The key model analyzed in depth is critical percolation on the hexagonal lattice. For this model, the critical exponents, previously determined using the now-famous Schramm-Loewner evolution, appear here in the study of sensitivity behaviour. Even for this relatively simple model, beyound the Fourier-analytic setup, there are three crucially important but distinct approaches: hypercontractivity of operators, connections to randomized algorithms, and viewing the spectrum as a random Cantor set. This book assumes a basic background in probability theory and integration theory. Each chapter ends with exercises, some straightforward, some challenging.” (publisher’s description).
The sections of the book are the following: Preface; Notations; Boolean functions and key concepts; Percolation in a nutshell; Sharp thresholds and the critical point for 2-d percolation; Fourier analysis of Boolean functions (first facts); Hypercontractivity and its applications; First evidence of noise sensitivity of percolation; Anomalous fluctuations; Randomized algorithms and noise sensitivity; The spectral sample; Sharp noise sensitivity of percolation; Applications to dynamical percolation; For the connoisseur; Further directions and open problems; References; Index.

MSC:
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
06E10 Chain conditions, complete algebras
94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
60H40 White noise theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
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