2020.07.02 Ron Peled - Fluctuations of random surfaces and concentration inequalities
Channel: One World Probability
Date: 2020-07-02

Random surfaces in statistical physics are commonly modeled by a real-valued function on a d-dimensional lattice, whose probability density penalizes nearest-neighbor fluctuations according to an interaction potential U. The case U(x)=x^2 is the well-studied lattice Gaussian free field, while one-dimensional random surfaces are equivalent to random walks. Our focus is on dimensions d≥2 and general U. Brascamp-Lieb-Lebowitz conjectured in 1975 that such random surfaces are localized in dimensions d≥3 under mild assumptions on U. Their work establishes the conjecture when U is uniformly convex (its second derivative is uniformly bounded from zero), as a consequence of the Brascamp-Lieb concentration inequality. To date, this remains the main case for which the conjecture is verified, with the result missing even when U(x) = x^4. We establish new concentration inequalities for log-concave distributions, extending the Brascamp-Lieb inequality, and use them to prove localization in many new cases, including the family U(x) = |x|^p with p⋗1. Further consequences regard the maximum height for a class of random surfaces discussed by Deuschel-Giacomin (2000).

The talk will be a gentle introduction to the model and the results. No prior knowledge of random surfaces or log-concave distributions will be assumed.
Joint work with Alexander Magazinov.