In this chapter, our goal is to collect some of the facts and theorems we have seen so far in order to conclude that percolation crossings are indeed noise sensitive. Recall from the “BKS” Theorem (Theorem 1.21) that it is enough for this purpose to prove that influences are “small” in the sense that ΣkIk(fn)2 goes to 0. If we want to use only what we have actually proved in this book, namely Proposition 5.6, then we need to demonstrate (5.6) in this proposition.

In the first section, we deal with a careful study of influences in the case of percolation crossings on the triangular lattice. Then, we treat the case of ℤ2, where conformal invariance is not known. Finally, we speculate to what “extent” percolation is noise sensitive.

This whole chapter should be considered somewhat of a “pause” in our program, where we take the time to summarize what we have achieved so far in our understanding of the noise sensitivity of percolation, and what remains to be done if one wishes to obtain the exact “noise sensitivity exponent” as well as the existence of exceptional times for dynamical percolation.

Bounds on influences for crossing events in critical percolation on the triangular lattice

6.1.1 Setup

Fix a, b > 0, let us consider some rectangle [0, a·n]×[0, b·n], and let Rn be the set of hexagons in T that intersect [0, a·n]×[0, b·n]. Let fn be the event that there is a left-to-right crossing event in Rn. (This is the same event as in Example 1.22 in Chapter 1, but with ℤ2 replaced by T.) By the RSW Theorem 2.1, we know that {fn} is nondegenerate. Conformal invariance tells us that E[fn] = ℙ[fn = 1] converges as n → ∞. This limit is given by the so-called Cardy's formula.

In this chapter, we explain how the notion of revealment for so-called randomized algorithms can in some cases yield direct information concerning the energy spectrum that may allow not only noise sensitivity results but even quantitative noise sensitivity results.

BKS and randomized algorithms

In the previous chapter, we explained how Theorem 1.21 together with bounds on the pivotal exponent for percolation yields noise sensitivity for percolation crossings. However, in (BKS99), a different approach was in fact used for showing noise sensitivity which, while still using Theorem 1.21, did not use these bounds on the critical exponent. In that approach, one sees the first appearance of randomized algorithms. In a nutshell, the authors showed that (1) if a monotone function is very uncorrelated with all majority functions, then it is noise sensitive (in a precise quantitative sense) and (2) percolation crossings are very uncorrelated with all majority functions. The latter is shown by constructing a certain algorithm which, due to the RSW Theorem 2.1, looks at very few bits but still looks at enough bits to be able to determine the output of the function. This approach is detailed in Section 12.13 in Chapter 12.

The revealment theorem

An algorithm for a Boolean function f is an algorithm A that queries (asks the values of) the bits one by one, where the decision of which bit to ask can be based on the values of the bits previously queried, and stops once f is determined (being determined means that f takes the same value no matter how the remaining bits are set).

A randomized algorithm for a Boolean function f is the same as above but auxiliary randomness may also be used to decide the next value queried (including for the first bit). [In computer science, the term randomized decision tree would be used for our notion of randomized algorithm, but we will not use this terminology.]

In this first chapter, we set the stage for the book by presenting many of its key concepts of the book and stating a number of important theorems that we prove here.

Boolean functions

Definition 1.1 A Boolean function is a function from the hypercube Ωn : = {−1, 1}n into either {−1, 1} or {0, 1}.

Ωn is endowed with the uniform measure ℙ = ℙn = (½δ−1 + ½δ1)⊗n and E denotes the corresponding expectation. Occasionally, Ωn will be endowed with the general product measure but in such cases the p is made explicit. Ep then denotes the corresponding expectation.

An element of Ωn is denoted by either ω or ωn and its n bits by x1, …, xn so that ω = (x1,…, xn).

For the range, we choose to work with {−1, 1} in some contexts and {0, 1} in others, and at some specific places we even relax the Boolean constraint (i.e., that the function takes only two possible values). In these cases (which are clearly identified), we consider instead real-valued functions f: Ωn → ℝ.

A Boolean function f is canonically identified with a subset Af of Ωn via Af : = {ω : f(ω) = 1}.

Remark Often, Boolean functions are defined on {0, 1}n rather than Ωn = {−1, 1}n. This does not make any fundamental difference but, as we see later, the choice of {−1, 1}n turns out to be more convenient when one wishes to apply Fourier analysis on the hypercube.

In this chapter, our goal is to extend the technology we used to prove the KKL Theorems on influences and the BKS Theorem on noise sensitivity in a slightly different context: the study of fluctuations in first-passage percolation.

The model of first-passage percolation

Let us first explain what the model is. Let 0 < a < b be two positive numbers. We define a random metric on the graph ℤd, d ≥ 2 as follows. Independently for each edge e ∈ Ed, fix its length τe to be a with probability 1/2 and b with probability 1/2. This is represented by a uniform configuration ω ∈ {−1, 1}Ed.

This procedure induces a well-defined (random) metric distω on ℤd in the usual fashion. For any vertices x, y ∈ ℤd, let

distω(x, y): = inf {Στei(ω)}.

paths γ = {e1, …, ek}

connecting x → y

Remark In greater generality, the lengths of the edges are i.i.d. nonnegative random variables, but here, following (BKS03), we restrict ourselves to the above uniform distribution on {a, b} to simplify the exposition; see (BR08) for an extension to more general laws.

One of the main goals in first-passage percolation is to understand the large-scale properties of this random metric space. For example, for any T ≥ 1, one may consider the (random) ball

Bω(x, T): = {y ∈ ℤd : distω(x, y) ≤ T}.

To understand the name first-passage percolation, one can think of this model as follows. Imagine that water is pumped in at vertex x, and that for each edge e, it takes τe(ω) units of time for the water to travel across the edge e. Then, Bω(x, T) represents the region of space that has been wetted by time T.

The purpose of this book is to present a self-contained theory of Boolean functions through the prism of statistical physics. The material presented here was initially designed as a set of lecture notes for the 2010 Clay summer school, and we decided to maintain the informal style which, we hope, will make this book more reader friendly.

Before going into Chapter 1, where precise definitions and statements are given, we wish to describe in an informal manner what this book is about. Our main companion through the whole book will be what one calls a Boolean function. This is simply a function of the following type:

f: {0, 1}n → {0, 1}.

Traditionally, the study of Boolean functions arises more naturally in theoretical computer science and combinatorics. In fact, 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 book (Chapters 1 to 5) is devoted to a description of some of the main achievements in this field. For example, a crucial result that has inspired much of the work presented here is the so-called KKL theorem (for Kahn–Kalai–Linial, 1989), which in essence says that any “reasonable” Boolean function has at least one variable that has a large influence on the outcome (namely at least Ω(logn/n)). See Theorem 1.14.

The second part of this book is devoted to the powerful use of Boolean functions in the context of statistical physics and in particular in percolation theory. It was recognized long ago that some of the striking properties that hold in great generality for Boolean functions have deep implications in statistical physics. For example, a version of the KKL theorem enables one to recover in an elegant manner the celebrated theorem of Kesten from 1980 that states that the critical point for percolation on ℤ2, pc(ℤ2), is 1/2.

In this section, we present a very natural model where percolation undergoes a time-evolution: this is the model of dynamical percolation described below. The study of the “dynamical” behavior of percolation as opposed to its “static” behavior turns out to be very rich: interesting phenomena arise especially at the phase transition point. We will see that in some sense, dynamical planar percolation at criticality is a very unstable or chaotic process. In order to understand this instability, sensitivity of percolation (and therefore its Fourier analysis) will play a key role. In fact, the original motivation for the paper (BKS99) on noise sensitivity was to solve a particular problem in the subject of dynamical percolation. (Ste09) provides a recent survey on the subject of dynamical percolation.

We mention that one can read all but the last section of the present chapter without having read Chapter 10.

The model of dynamical percolation

This model was introduced by Häaggströom, Peres, and Steif (HPS97) inspired by a question that Paul Malliavin asked at a lecture at the Mittag-Leffler Institute in 1995. This model was invented independently by Itai Benjamini.

In the general version of this model as it was introduced, given an arbitrary graph G and a parameter p, the edges of G switch back and forth according to independent two-state continuous time Markov chains where closed switches to open at rate p and open switches to closed at rate 1 − p. Clearly, the product measure with density p, denoted by πp in this chapter, is the unique stationary distribution for this Markov process. The general question studied in dynamical percolation is whether, when we start with the stationary distribution πp, there exist atypical times at which the percolation structure looks markedly different than that at a fixed time.

We end this book with a concise list of open questions which are grouped by theme.

Randomized algorithms

13.1.1 Best randomized algorithm for Iterated Majority

It may look surprising (owing to the simple iterative structure of the Iterated Majority function introduced in Example 1.5), but the following problem is to our knowledge not settled.

Open Problem 13.1Find the randomized algorithm with smallest possible revealment for the Iterated Majority function fk on n = 3k bits. Even the order of magnitude of the decay to 0 (i.e., the exponent) is not known although there are some known bounds.

13.1.2 Best randomized algorithms for critical percolation

Open Problem 13.2

(a) Find algorithms for crossing events in percolation on the triangular lattice that examine on average at most n7/4−∈ sites for some ∈>0.

(b) Show that any algorithm examines on average at least n6/4+∈ sites for some ∈ > 0. (This would strengthen the lower bound obtained in Section 8.4 in Chapter 8).

13.1.3 Randomized algorithm and spectrum One way to obtain better bounds on the noise sensitivity of a Boolean function through the randomized algorithm approach (e.g., the percolation crossing events fn) is to find the algorithms with the smallest possible revealments. Another possible direction is to address the following problem.

Open Problem 13.3Prove a sharper version of Theorem 8.2 from (SS10).

13.1.4 Randomized algorithms and pivotal points

In (PSSW07), the authors introduce a seemingly very efficient randomized algorithm: one starts at some random initial site (using a well-chosen initial distribution) and then at each step, one picks the site that has the largest probability to be pivotal conditioned on what has been revealed so far. See (PSSW07). Their numerical simulations suggest that this randomized algorithm is more efficient than the one we used in Chapter 8 using an interface. Yet, nothing is rigorously known.