Upper bounds on the fluctuations for a class of degenerate convex ∇ϕ-interface models

We derive upper bounds on the fluctuations of a class of random surfaces of the ∇ϕ-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 ≥ 3. The proof makes use of the Helffer-Sjöstrand representation formula (originally introduced by Helffer and Sjöstrand (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)).