diffusion entropy and the geometry of small sets

diffusion, entropy, and the geometry of small sets James R. Lee University of Washington Joint work with Ronen Eldan

a prelude

noise and smoothness

noise and smoothness Many applications: PCPs & hardness of approximation, statistical physics, threshold phenomena, social choice, circuit complexity,

noise and smoothness Hypercontractive inequality [Bonami, Gross, Nelson]:

noise and smoothness Hypercontractive inequality [Bonami, Gross, Nelson]:

noise and smoothness Hypercontractive inequality [Bonami, Gross, Nelson]:

the convolution conjecture

anti-concentration of temperature Convolution conjecture [Talagrand 1989]: Equivalent to the conjecture that “heat” cannot concentrate near a single (high) “temperature”

the Gaussian case

the Gaussian case

the Gaussian case Ornstein-Uhlenbeck semi-group:

the Gaussian case Theorem [Eldan-L 2014]:

the Gaussian case Theorem [Eldan-L 2014]:

the Gaussian case Theorem [Eldan-L 2014]:

some difficulties Good: noise insensitive Bad: boundary far from origin half space dust

proof ideas arguing about small-probability events is tough, but remember the prelude… random measure

conditioning

Explicit form: Theorem [Lehec 2010]:

an optimal coupling is a martingale

proof sketch

proof sketch

Girsanov’s theorem Then under the change of measure:

the masochistic part

the masochistic part

conclusion

Peres-Tetali conjecture