Towards a Nonperturbative Renormalization Group Analysis

We prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field. As a consequence we establish a certain asymptotic formula for the partition function. In some ways, this is a realization of Gawedzki and Kupiainen’s idea to use correlation inequalities to augment the rigorous renormalization group methods. We use the more particular suggestion of Spencer to use certain inequalities of Brascamp and Lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by Helffer and Sjöstrand.