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We derive upper bounds on the fluctuations of a class of random surfaces ofthe $\nabla \phi$-type with convex interaction potentials. The Brascamp-Liebconcentration inequality provides an upper bound on these fluctuations foruniformly convex potentials. We extend these results to twice continuouslydifferentiable convex potentials whose second derivative grows asymptoticallylike a polynomial and may vanish on an (arbitrarily large) interval.Specifically, we prove that, when the underlying graph is the $d$-dimensionaltorus 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 Naddafand Spencer (1997) and Giacomin, Olla Spohn (2001) to identify the scalinglimit of the model), the anchored Nash inequality (and the correspondingon-diagonal heat kernel upper bound) established by Mourrat and Otto (2016) andEfron's monotonicity theorem for log-concave measures (Efron (1965)).

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