Confinement of unimodal probability distributions and an FKG-Gaussian correlation inequality

While unimodal probability distributions are well understood in dimension 1, the same cannot be said in high dimension without imposing stronger conditions such as log-concavity. I will explain a new approach to proving confinement (e.g., variance upper bounds) for high-dimensional unimodal distributions which are not log-concave, based on an extension of Royen's celebrated Gaussian correlation inequality. We will see how it yields new localization results for Ginzburg–Landau random surfaces with very general monotone potentials.

Time permitting, I will also mention a related result on the effective mass of the Fröhlich Polaron, which is joint work with Rodrigo Bazaes, Chiranjib Mukherjee, and S.R.S. Varadhan.