Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

We consider a non-interacting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Otto, West-dickenberg, and Villani from the quadratic to the general case. Using an asymmetric Brascamp-Lieb type inequality for covariances we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cramèr theorem.