Chapter 11: Gaussian space and Invariance Principles

The final destination of this chapter is a proof of the following theorem due to Mossel, O’Donnell, and Oleszkiewicz [MOO05a,MOO10], first mentioned in Chapter 5.25:

Majority Is Stablest Theorem Fix $\rho \in (0,1)$. Let $f : \{-1,1\}^n \to [-1,1]$ have $\mathop{\bf E}[f] = 0$. Then, assuming $\mathbf{MaxInf}[f] \leq \epsilon$, or more generally that $f$ has no $(\epsilon,\epsilon)$-notable coordinates, \[ \mathbf{Stab}_\rho[f] \leq 1 – \tfrac{2}{\pi} \arccos \rho + o_\epsilon(1). \]

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Chapter 10 notes

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Chapter 10 exercises

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§10.5: Highlight: General sharp threshold theorems

In Chapter 8.4 we described the problem of “threshold phenomena” for monotone functions $f : \{-1,1\}^n \to \{-1,1\}$.
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§10.4: More on randomization/symmetrization

In Section 3 we collected a number of consequences of the General Hypercontractivity Theorem for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$. All of these had a dependence on “$\lambda$”, the least probability of an outcome under $\pi$. This can sometimes be quite expensive; for example, the KKL Theorem and its consequence Theorem 28 are trivialized when $\lambda = 1/n^{\Theta(1)}$.
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§10.3: Applications of general hypercontractivity

In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5.
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§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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Chapter 10: Advanced hypercontractivity

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:

The General Hypercontractivity Theorem Let $(\Omega_1, \pi_1), \dots, (\Omega_n, \pi_n)$ be finite probability spaces, in each of which every outcome has probability at least $\lambda$. Let $f \in L^2(\Omega_1 \times \cdots \times \Omega_n, \pi_1 \otimes \cdots \otimes \pi_n)$. Then for any $q > 2$ and $0 \leq \rho \leq \frac{1}{\sqrt{q-1}} \cdot \lambda^{1/2-1/q}$, \[ \|\mathrm{T}_\rho f\|_q \leq \|f\|_2 \quad\text{and}\quad \|\mathrm{T}_\rho f\|_2 \leq \|f\|_{q'}. \] (In fact, the upper bound on $\rho$ can be slightly relaxed to the value stated in Theorem 17 of this chapter.)

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Chapter 9 notes

The history of the Hypercontractivity Theorem is complicated.
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