Noise Sensitivity of Boolean Functions and Percolation: (Series Number 5 Institute of Mathematical Statistics Textbooks)

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.

Table of Contents:
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.

About the Author :
Christophe Garban is a professor of mathematics at Université Lyon I, France. Jeffrey Steif is a Professor of Mathematical Sciences at Chalmers University of Technology, Gothenburg, Sweden.

Review :
'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 'This book is about a beautiful mathematical story, centered around the wonderful, ever-changing theory of probability and rooted in questions of physics and computer science. Christophe Garban and Jeffrey Steif, both heroes of the research advances described in the book, tell the story and lucidly explain the underlying probability theory, combinatorics, analysis, and geometry - from a very basic to a state-of-the-art level. The authors make great choices on what to explain and include in the book, leaving the readers with perfect conceptual understanding and technical tools to go beyond the text and, at the same time, with much appetite for learning and exploring even further.' Gil Kalai, Hebrew University 'Boolean functions map many bits to a single bit. Percolation is the study of random configurations in the lattice and their connectivity properties. These topics seem almost disjointed - except that the existence of a left-to-right crossing of a square in the 2D lattice is a Boolean function of the edge variables. This observation is the beginning of a magical theory, developed by Oded Schramm and his collaborators, in particular Itai Benjamini, Gil Kalai, Gabor Pete, and the authors of this wonderful book. The book expertly conveys the excitement of the topic; connections with discrete Fourier analysis, hypercontractivity, randomized algorithms, dynamical percolation, and more are explained rigorously, yet without excessive formality. Numerous open problems point the way to the future.' Yuval Peres, Principal Researcher, Microsoft 'Without hesitation, I can recommend this monograph to any probabilist who has considered venturing into the domain of noise sensitivity of Boolean functions. All fundamental concepts of the field such as influence or noise sensitivity are explained in a refreshingly accessible way, so that only a minimal understanding of probability theory is assumed. The authors succeed in guiding the reader gently from the basics to the most recent seminal developments in Fourier analysis of Boolean functions, familiarizing her or him with all the modern machinery along the way.' Christian Hirsch, Mathematical Reviews '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