Remarks on the Discrete Cube
Talkat Probability, Geometry, and Computation in High Dimensions Boot Camp (Simons Institute, Fall 2020) · 2020-08-19..08-28
summary
Provides a self-contained proof that Var(f) ≤ Σi E[(Di f)^2] with optimal constant 1 by building an interpolation by continuous-time random walk (independent Poisson clocks per coordinate) and analyzing its Markov semigroup. Derives d/dt P_t f = ΔP_t f with Δ = −Σi Di, shows derivative forgets initial data at rate e^(−2t), and recovers Efron–Stein as a special case. Promises later extensions to vector-valued inequalities and elementary proofs avoiding Fourier.
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