In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9:
Continue reading §10.1: The Hypercontractivity Theorem for uniform $\pm 1$ bits


In this section we’ll prove the full Hypercontractivity Theorem for uniform $\pm 1$ bits stated at the beginning of Chapter 9: In this chapter we complete the proof of the Hypercontractivity Theorem for uniform $\pm 1$ bits. We then generalize the $(p,2)$ and $(2,q)$ statements to the setting of arbitrary product probability spaces, proving the following:
Continue reading Chapter 10: Advanced hypercontractivity The history of the Hypercontractivity Theorem is complicated. Recalling the social choice setting of Chapter 2.5, consider a $2$candidate, $n$voter election using a monotone voting rule $f : \{1,1\}^n \to \{1,1\}$. We assume the impartial culture assumption (that the votes are independent and uniformly random), but with a twist: one of the candidates, say $b \in \{1,1\}$, is able to secretly bribe $k$ voters, fixing their votes to $b$. (Since $f$ is monotone, this is always the optimal way for the candidate to fix the bribed votes.) How much can this influence the outcome of the election? With the $(2,q)$ and $(p,2)$Hypercontractivity Theorems in hand, let’s revisit some applications we saw in Sections 1 and 2. At this point we have established that if $f : \{1,1\} \to {\mathbb R}$ then for any $p \leq 2 \leq q$, \[ \\mathrm{T}_{\sqrt{p1}} f\_2 \leq \f\_p, \qquad \\mathrm{T}_{1/\sqrt{q1}} f\_q \leq \f\_2. \] We would like to extend these facts to the case of general $f : \{1,1\}^n \to {\mathbb R}$; i.e., establish the $(p,2)$ and $(2,q)$Hypercontractivity Theorems stated at the beginning of the chapter. A natural approach is induction. Although you can get a lot of mileage out of studying the $4$norm of random variables, it’s also natural to consider other norms. An immediate consequence of the Bonami Lemma is that for any $f : \{1,1\}^n \to {\mathbb R}$ and $k \in {\mathbb N}$, \begin{equation} \label{eqn:24hyperconk} \\mathrm{T}_{1/\sqrt{3}} f^{=k}\_4 = \tfrac{1}{\sqrt{3}^k} \f^{=k}\_4 \leq \f^{=k}\_2. \end{equation} As anyone who has worked in probability knows, a random variable can sometimes behave in rather “unreasonable” ways. It may be never close to its expectation. It might exceed its expectation almost always, or almost never. It might have finite $1$st, $2$nd, and $3$rd moments, but an infinite $4$th moment. All of this poor behaviour can cause a lot of trouble — wouldn’t it be nice to have a class of “reasonable” random variables? 

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