Finite Range Decomposition for Gaussian Measures with Improved Regularity

TL;DR

This paper improves finite range decompositions of Gaussian measures on tori, controlling Fourier decay and regularity, which benefits renormalization group methods in statistical mechanics.

Contribution

It enhances previous finite range decomposition results by controlling Fourier decay and regularity, avoiding regularity loss in renormalization group applications.

Findings

Controlled decay behavior of kernels in Fourier space.

Established regularity of the convolution map with finite range measures.

Avoided regularity loss in anisotropic renormalization group problems.

Abstract

We consider a family of gradient Gaussian vector fields on the torus $(\mathbb{Z}/L^N\mathbb{Z})^d$. Adams, Koteck\'{y}, M\"{u}ller and independently Bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators, i.e., the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter $L^k$. We improve this result and show that the decay behaviour of the kernels in Fourier space can be controlled. Then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition. In particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics.