Log Concavity and Concentration of Measure on the Discrete Hypercube
summary
Defines a usable discrete curvature: a density on {-1,1}^n is beta-semi log-concave if the multilinear (harmonic) extension of its log has Hessian bounded above by beta I, and this implies a nontrivial variance bound: for every 1‑Lipschitz f Var_nu(f) ≤ n^{2−c/beta} for a universal c. Proof reduces concentration to a transport estimate—small Wasserstein distance between tilts—and constructs a Brownian‑driven SDE coupling using the Lipschitz vector field a_nu (log‑Laplace gradient) to control transports. Quadratic case: Poincaré holds up to operator norm 1/4.
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