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.
In The Press
'Presented in an orderly, accessible manner, this book provides an excellent exposition of the general theory of noise sensitivity and its beautiful and deep manifestation in two dimensional critical percolation. The authors, both of whom are major contributors to the theory, have produced a very thoughtful work, bringing the intuition and motivations first. Noise sensitivity is a natural concept that recently found diverse applications, ranging from quantum computation and complexity theory to statistical physics and social choice. Two dimensional critical percolation is a striking and canonical random object. The book elegantly unfolds the story of integrating the general theory of noise sensitivity into a concrete study, allowing for a new understanding of the percolation process.' Itai Benjamini, Weizmann Institute of Science, Israel