On A Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus

Abstract

In previous papers, [M1, M2], [M3], we proved the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha }(x-y,m^2)=((-\Delta {)}^{\frac{\alpha }{2}}+m^2{)}^{-1}(x-y)$, for $0<\alpha <2$ and all real $m$, in the lattice ${\mathbf{Z}}^d$ for dimension $d\ge 2$. In this paper, which is a continuation of the previous one, we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus ${\mathbf{Z}}^d/L^{N+1}{\mathbf{Z}}^d$ for $d\ge 2$ provided $m\ne 0$ and $0<\alpha <2$. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter $m^2$. Here $L$ is any odd positive integer and $N\ge 2$ is any positive integer.

17 pages

References in corpus (2)

• Critical exponents for long-range O(n) models below the upper critical dimension
• A renormalisation group method. III. Perturbative analysis

Cited by in corpus (3)

• Introduction to a renormalisation group method
• Critical exponents for long-range O(n) models below the upper critical dimension
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension