Abstract: 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, relying on an FKG-based 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 discuss a related result on the effective mass of the Fröhlich Polaron, obtained with Rodrigo Bazaes, Chiranjib Mukherjee, and S.R.S. Varadhan. (Note that I gave a Rhein-Main talk here last year on the Polaron, but the new method works quite differently.)