Talagrand's convolution conjecture and geometry via coupling
Consider an image with two colors--black and white--and where only 1% of the pixels are white. If we apply a Gaussian blur, can it be that the non-black pixels of the (now greyscale) image are largely concentrated on a single shade of grey? Sure, if we apply enough blur, the whole picture will be a dull grey. But can this happen for a very light shade of grey? That seems preposterous--certainly the colors should fade gracefully from light to dark. Talagrand conjectured in 1989 that we should see at least 50 shades of grey. We will prove it.
More formally: It is a well-known phenomenon that functions on Gaussian space become smoother under the Ornstein-Uhlenbeck semigroup. Versions of this have proved to have many powerful applications. For instance, Nelson's hypercontractive inequality shows that if \(p > 1\), then \(L^p\) functions are sent to \(L^q\) functions for some \(q > p\). In 1989, Talagrand conjectured that quantitative smoothing is achieved even for functions which are only \(L^1\), in the sense that under the semigroup, such functions have tails that are strictly better than those predicted by Markov's inequality and preservation of mass. Ball, Barthe, Bednorz, Oleszkiewicz, and Wolff (2010) proved that this holds in fixed dimension. We resolve Talagrand's conjecture conjecture positively (with no dimension dependence).
The key insight will be a method of stalking the white pixels while appearing as nonchalant as possible. (This will be cast as a problem in stochastic control theory.)
This is joint work with Ronen Eldan.