§11.1: Gaussian space and the Gaussian noise operator

We begin with a few definitions concerning Gaussian space.
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The book is available for pre-order

I’m happy to announce that the book is very nearly completed. In fact, you can pre-order a copy now, either directly from Cambridge University Press, or from Amazon (currently with a 10% discount). If all goes well, the book will become physically available at the end of May. Fairly soon thereafter I will also make a pdf “final draft” version freely available here at the website.

In the meantime, I will continue to serialize the final chapter (Chapter 11) as usual over the next month. In fact, I had to trim and glue together the planned Chapters 11 and 12. Also, as mentioned earlier, I dropped a chapter on Additive Combinatorics that would have gone between Chapters 8 and 9, although its “highlight”, Sanders’s Theorem, is here on the blog. Oh well, I had to wrap this project up at some point; at least you’ll have an idea of what will be in the 2nd Edition. :)

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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