Fluctuation estimates for sub-quadratic gradient field actions
David Brydges, Thomas Spencer · Journal of Mathematical Physics · 2012
In this article we estimate fluctuations of the scalar field ϕ for a special class of sub-quadratic actions which grow like |∇ϕ|2α, 0 < α < 1. In particular if α = 1/2 we show that in three dimensions \documentclass[12pt]{minimal}\begin{document}$\langle e^{\gamma \phi _0}\rangle$\end{document}⟨eγϕ0⟩ is bounded for γ small. For each edge (jk) we introduce an auxiliary field \documentclass[12pt]{minimal}\begin{document}$t_{jk} \in \mathbb R$\end{document}tjk∈R to express the action as a superposition of Gaussian free fields. The effective action which arises from integrating over the Gaussian field is shown to be convex in t. The Brascamp-Lieb inequality is then applied to obtain the desired estimates on a nonuniformly elliptic Green's function.