Log-concave Polynomials and Spectral Independence 3

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" summary

Shows that spectral independence of all conditionals gives a concrete mixing bound: with ζ-SI the Glauber spectral gap is at least 1/n^{1+ζ}; adding a global Markov property on bounded-degree graphs and bounded marginals yields a log Sobolev constant Ω(1/n) and hence n log n mixing. Proof tools are variance/entropy factorization via Dirichlet forms, an induction that peels off coordinates, and a shattering into small components lemma (random deletion on Δ-bounded graphs yields exponentially small large-component tails) that makes the factorization effective.

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" chapters (10)

1. 0:00Introduction and concepts
2. 1:57Mixing theorems
3. 6:25Analysis for product measures
4. 10:41Inductive factorization
5. 13:31Restating the theorem
6. 18:57Proof sketch of the key lemma
7. 29:11Dynamical perspective
8. 36:02Conditional independence
9. 40:05Shattering bounds
10. 43:23Model dependencies and closing

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