Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality

The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on ℤ^d when d≥ 3 for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials U:ℝ→ℝ satisfying U'(x)≥min(ε x,1+ε/x) on x∈ℝ^+. Given correspondingly stronger growth conditions on U, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.