Upper bounds on the fluctuations for a class of degenerate convex $\nabla \phi$-interface models

We derive upper bounds on the fluctuations of a class of random surfaces of the $\nabla \phi$-type with convex interaction potentials. The Brascamp-Lieb concentration inequality provides an upper bound on these fluctuations for uniformly convex potentials. We extend these results to twice continuously differentiable convex potentials whose second derivative grows asymptotically like a polynomial and may vanish on an (arbitrarily large) interval. Specifically, we prove that, when the underlying graph is the $d$-dimensional torus of side length $L$, the variance of the height is smaller than $C \ln L$ in two dimensions and remains bounded in dimension $d \geq 3$. The proof makes use of the Helffer-Sj\"{o}strand representation formula (originally introduced by Helffer and Sj\"{o}strand (1994) and used by Naddaf and Spencer (1997) and Giacomin, Olla Spohn (2001) to identify the scaling limit of the model), the anchored Nash inequality (and the corresponding on-diagonal heat kernel upper bound) established by Mourrat and Otto (2016) and Efron's monotonicity theorem for log-concave measures (Efron (1965)).