Noise Sensitivity of Boolean Functions and Percolation

This is the first book to cover the theory of noise sensitivity of Boolean functions with particular emphasis on critical percolation.

Christophe Garban (Author), Jeffrey E. Steif (Author)

9781107432550, Cambridge University Press

Paperback / softback, published 22 December 2014

222 pages, 29 b/w illus. 75 exercises
23 x 15.1 x 1.4 cm, 0.33 kg

'Considerable effort was made to make the book as thorough and concise as possible but still readable and friendly. … It is clear that it will turn out to be the 'go to' source for studying the subject of noise sensitivity of Boolean functions.' Eviatar B. Procaccia, Bulletin of the American Mathematical Society

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.

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.

Subject Areas: Algorithms & data structures [UMB], Statistical physics [PHS], Probability & statistics [PBT], Calculus & mathematical analysis [PBK], Discrete mathematics [PBD]