Influences in Gaussian Space
summary
Defines a new geometric influence by integrating one-dimensional boundary (lower Minkowski) measures of coordinate fibers, shows it equals an L∞ surface boundary for monotone sets, and proves a Gaussian KKL: some coordinate has influence at least of order 1/√n·√(log n) (sharp for semi-infinite boxes). Extends to Boltzmann measures, gives a Gaussian Russo-type formula, a Talagrand-style correlation bound, BKS noise-sensitivity analog via CLT, and a semigroup approach using Ornstein–Uhlenbeck smoothing, hypercontractivity, and 1D isoperimetry.
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chapters (10)
- 0:00Introduction to influences
- 1:27Discrete space review
- 4:50Threshold phenomena
- 8:33Defining geometric influence
- 12:33KKL theorem in Gaussian space
- 17:58Talagrand variance bounds
- 22:35Markov semigroup framework
- 27:52Thresholds and smoothness
- 32:46Noise sensitivity
- 39:17Limits and applications
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