Quantitative Correlation Inequalities via Semigroup Interpolation
summary
Proves a quantitative Gaussian correlation bound: for symmetric convex sets K,L the correlation ≥ φ(Σ_i Inf_i(K)·Inf_i(L)) with Inf_i defined as the negated degree‑2 Hermite coefficient and φ(x)=x / log^2(1/x). Proof interpolates expectations via the Ornstein‑Uhlenbeck operator, expands in the Hermite basis to get a power series, then applies a complex‑analytic lemma based on the Hadamard three circles theorem; method recovers Talagrand‑style bounds on other domains up to an extra log factor.
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chapters (22)
- 0:00Intro
- 0:38Example 1: Harris-Kleitman Inequality (1966)
- 1:09Example 2: Royen's Inequality (2014)
- 2:11Towards Quantitative Inequalities
- 2:41Revisiting Harris-Kleitman
- 4:34Influences of Variables on Boolean Functions
- 5:19Robustifying Harris-Kleitman
- 6:03Talagrand's Correlation Inequality (1996)
- 6:48Quantitative Correlation Inequalities
- 8:39Proof Strategy
- 9:55Outline
- 10:21Harmonic Analysis over Gaussian Space
- 10:56The Hermite Basis
- 12:26The Ornstein-Uhlenbeck Noise Operator
- 13:35Hermite Expansion under the OU Operator
- 14:32Quick Recap
- 17:29A Quantitative GCI
- 17:53Proof Sketch
- 21:29An Extremal Bound for Complex Power Series
- 27:51Influences for Symmetric, Convex Sets
- 29:05Generalizing to Other Domains
- 31:06Summary
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