We begin with a few definitions concerning Gaussian space.
[...]


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 $f$ [...] [...] [...] In Chapter 8.4 we described the problem of “threshold phenomena” for monotone functions $f : \{1,1\}^n \to \{1,1\}$. [...] 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 [...] In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5. [...] 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. [...] 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: The General Hypercontractivity Theorem Let $(\Omega_1, \pi_1), \dots, (\Omega_n, \pi_n)$ be finite probability spaces, in each of which [...] 

Copyright © 2018 Ryan O'Donnell  All Rights Reserved 
Recent comments