Lectures on noise sensitivity and percolation

The goal of this set of lectures is to combine two seemingly unrelated topics: • The study of Boolean functions, a field particularly active in computer science • Some models in statistical physics, mostly percolation The link between these two fields can be loosely explained as follows: a percolation configuration is built out of a collection of i.i.d " bits " which determines whether the corresponding edges, sites, or blocks are present or absent. In that respect, any event concerning percolation can be seen as a Boolean function whose input are precisely these " bits ". Over the last 20 years, mainly thanks to the computer science community, a very rich structure has emerged concerning the properties of Boolean functions. The first part of this course will be devoted to a description of some of the main achievements in this field. In some sense one can say, although this is an exaggeration, that computer scientists are mostly interested in the stability or robustness of Boolean functions. As we will see later in this course, the Boolean functions which " encode " large scale properties of critical percolation will turn out to be very sensitive to small perturbations. This phenomenon corresponds to what we will call noise sensitivity. Hence, the Boolean functions one wishes to describe here are in some sense orthogonal to the Boolean functions one encounters, ideally, in computer science. Remarkably, it turns out that the tools developed by the computer science community to capture the properties and stability of Boolean functions are also suitable for the study of noise sensitive functions. This is why it is worth us first spending some time on the general properties of Boolean functions. One of the main tools needed to understand properties of Boolean functions is Fourier analysis on the hypercube. Noise sensitivity will correspond to our Boolean function being of " high frequency " while stability will correspond to our Boolean function being of " low frequency ". We will apply these ideas to some other models from statistical mechanics as well; namely, first passage percolation and dynamical percola-tion. Some of the different topics here can be found (in a more condensed form) in [Gar10]. 5 Acknowledgements We wish to warmly thank the organizers David Ellwood, Charles Newman, Vladas Sidoravicius and Wendelin Werner for inviting us to give this course at the Clay summer school 2010 in Buzios. It was a …