Remarks on the Discrete Cube

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Boolean functions on the discrete cube beat the usual Poincaré bound by two mechanisms: replace L2 by an L1 Poincaré inequality to remove a square, and exploit much faster semigroup decay for low-influence functions. The quantitative claim is that for f:{±1}^n→{0,1} equilibrium arrives at time T = O(1 / log(1 / Σ_i Inf_i(f)^2), so Var(P_T f) ≤ ½ Var(f), yielding Talagrand‑style bounds (extra log n factor on max influence). Proof uses semigroup interpolation, an AM–GM reverse trick, tensorization and the Efron–Stein martingale. The tribes construction attains the extremal behavior.

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" chapters (8)

1. 0:00Start
2. 4:22Interpolation Method
3. 8:57Improving the Poncare Inequality
4. 11:04The Poincare Inequality
5. 18:02The Tribes Function
6. 56:28The Amgm Principle
7. 1:04:45Level One and Level Two Inequalities
8. 1:10:57Description of the Alternate Group