# Noise Sensitivity of Boolean Functions and Percolation (Institute of Mathematical Statistics Textbooks)

Cambridge University Press, 12/22/2014

EAN 9781107432550, ISBN10: 1107432553

Paperback, 222 pages, 22.9 x 15.2 x 1.3 cm

Language: English

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 SchrammÃ¢â‚¬â€œLoewner 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.