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Cover: 9781107432550 | Noise Sensitivity of Boolean Functions and Percolation | Steif | Buch
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Noise Sensitivity of Boolean Functions and Percolation
Taschenbuch von Christophe Garban (u. a.)
Sprache: Englisch

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Beschreibung
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 behavior. Even for this relatively simple model, beyond the Fourier-analytic set-up, 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.
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 behavior. Even for this relatively simple model, beyond the Fourier-analytic set-up, 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.
Über den Autor
Christophe Garban is a professor of mathematics at Université Lyon I, France.
Zusammenfassung
This account of the new and exciting area of noise sensitivity of Boolean functions - in particular applied to critical percolation - is designed for graduate students and researchers in probability theory, discrete mathematics, and theoretical computer science. It assumes a basic background in probability theory and integration theory. Each chapter ends with exercises.
Inhaltsverzeichnis
1. Boolean functions and key concepts; 2. Percolation in a nutshell; 3. Sharp thresholds and the critical point; 4. Fourier analysis of Boolean functions; 5. Hypercontractivity and its applications; 6. First evidence of noise sensitivity of percolation; 7. Anomalous fluctuations; 8. Randomized algorithms and noise sensitivity; 9. The spectral sample; 10. Sharp noise sensitivity of percolation; 11. Applications to dynamical percolation; 12. For the connoisseur; 13. Further directions and open problems.
Details
Erscheinungsjahr: 2015
Fachbereich: Wahrscheinlichkeitstheorie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 222
ISBN-13: 9781107432550
ISBN-10: 1107432553
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Garban, Christophe
Steif, Jeffrey E.
Hersteller: Cambridge University Press
Maße: 229 x 152 x 13 mm
Von/Mit: Christophe Garban (u. a.)
Erscheinungsdatum: 05.03.2015
Gewicht: 0,329 kg
preigu-id: 105198082
Über den Autor
Christophe Garban is a professor of mathematics at Université Lyon I, France.
Zusammenfassung
This account of the new and exciting area of noise sensitivity of Boolean functions - in particular applied to critical percolation - is designed for graduate students and researchers in probability theory, discrete mathematics, and theoretical computer science. It assumes a basic background in probability theory and integration theory. Each chapter ends with exercises.
Inhaltsverzeichnis
1. Boolean functions and key concepts; 2. Percolation in a nutshell; 3. Sharp thresholds and the critical point; 4. Fourier analysis of Boolean functions; 5. Hypercontractivity and its applications; 6. First evidence of noise sensitivity of percolation; 7. Anomalous fluctuations; 8. Randomized algorithms and noise sensitivity; 9. The spectral sample; 10. Sharp noise sensitivity of percolation; 11. Applications to dynamical percolation; 12. For the connoisseur; 13. Further directions and open problems.
Details
Erscheinungsjahr: 2015
Fachbereich: Wahrscheinlichkeitstheorie
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 222
ISBN-13: 9781107432550
ISBN-10: 1107432553
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Garban, Christophe
Steif, Jeffrey E.
Hersteller: Cambridge University Press
Maße: 229 x 152 x 13 mm
Von/Mit: Christophe Garban (u. a.)
Erscheinungsdatum: 05.03.2015
Gewicht: 0,329 kg
preigu-id: 105198082
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