§10.2: Hypercontractivity of general random variables

Let’s now study hypercontractivity for general random variables. By the end of this section we will have proved the General Hypercontractivity Theorem stated at the beginning of the chapter.

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§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:

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§9.6: Highlight: The Kahn–Kalai–Linial Theorem

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 $k$ [...]

§9.5: Applications of hypercontractivity

With the $(2,q)$- and $(p,2)$-Hypercontractivity Theorems in hand, let’s revisit some applications we saw in Sections 1 and 2.

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§9.4: Two-function hypercontractivity and induction

At this point we have established that if $f : \{-1,1\} \to {\mathbb R}$ then for any $p \leq 2 \leq q$, \[ \|\mathrm{T}_{\sqrt{p-1}} f\|_2 \leq \|f\|_p, \qquad \|\mathrm{T}_{1/\sqrt{q-1}} f\|_q \leq \|f\|_2. \] We would like to extend these facts to the case of general $f : \{-1,1\}^n \to {\mathbb R}$; i.e., establish the $(p,2)$- [...]

§9.3: $(2,q)$- and $(p,2)$-hypercontractivity for a single bit

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.

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§9.2: Small subsets of the hypercube are noise-sensitive

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:2-4-hypercon-k} \|\mathrm{T}_{1/\sqrt{3}} f^{=k}\|_4 = \tfrac{1}{\sqrt{3}^k} \|f^{=k}\|_4 \leq \|f^{=k}\|_2. \end{equation}

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§9.1: Low-degree polynomials are reasonable

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 [...]

§8.6: Highlight: Randomized decision tree complexity

A decision tree $T$ for $f : \{-1,1\}^n \to \{-1,1\}$ can be thought of as a deterministic algorithm which, given adaptive query access to the bits of an unknown string $x \in \{-1,1\}^n$, outputs $f(x)$. E.g., to describe a natural decision tree for $f = \mathrm{Maj}_3$ in words: “Query $x_1$, then $x_2$. If they [...]

§8.5: Abelian groups

The previous section covered the case of $f \in L^2(\Omega^n, \pi^{\otimes n})$ with $|\Omega| = 2$; there, we saw it could be helpful to look at explicit Fourier bases. When $|\Omega| \geq 3$ this is often not helpful, especially if the only “operation” on the domain is equality. For example, if $f : \{\mathsf{Red}, [...]