Finite Range Decomposition for Gaussian Measures with Improved Regularity

Abstract

We consider a family of gradient Gaussian vector fields on the torus $(\mathbb{Z}/L^N\mathbb{Z}{)}^d$. Adams, Kotecký, Müller 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.

28 pages, more general interactions and smoothness statement, construction now based on the Bauerschmidt decomposition

References in corpus (7)

• Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis
• Scaling limits and critical behaviour of the 4-dimensional n-component $|\phi {|}^4$ spin model
• A renormalisation group method. III. Perturbative analysis
• A simple method for finite range decomposition of quadratic forms and Gaussian fields
• A renormalisation group method. V. A single renormalisation group step
• Strict Convexity of the Surface Tension for Non-convex Potentials
• Finite Range Decomposition for Gaussian Measures with Improved Regularity

Cited by in corpus (5)

• Introduction to a renormalisation group method
• Cauchy-Born Rule from Microscopic Models with Non-convex Potentials
• Finite Range Decomposition for Gaussian Measures with Improved Regularity
• Scaling limit and strict convexity of free energy for gradient models with non-convex potential
• Decay of covariance for gradient models with non-convex potential