  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 $\mathbb Z^d$ when $d\geq 3$ for very general
potentials, matching the behavior of the Gaussian free field. We confirm this
behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying
$U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$.
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.
