On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian

TL;DR

This paper establishes a finite range decomposition for the resolvent of a fractional Laplacian operator, applicable in lattice and continuum models, aiding the analysis of long-range interactions in statistical physics and field theory.

Contribution

It proves the existence and regularity of a finite range decomposition for the fractional Laplacian resolvent in both lattice and continuum settings, extending previous results for the case when mass parameter is zero.

Findings

Decomposition exists for all real mass parameters in dimensions d≥2.

Applicable to models with long-range jumps and interactions.

Enhances tools for analyzing critical and off-critical renormalization group flows.

Abstract

We prove the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha<2$ and all real $m$, in the lattice ${\mathbf Z}^{d}$ as well as in the continuum ${\mathbf R}^{d}$ for dimension $d\ge 2$. This resolvent occurs as the covariance of the Gaussian measure underlying weakly self- avoiding walks with long range jumps (stable L\'evy walks) as well as continuous spin ferromagnets with long range interactions in the long wavelength or field theoretic approximation. The finite range decomposition should be useful for the rigorous analysis of both critical and off-critical renormalisation group trajectories. The decomposition for the special case $m=0$ was known and used earlier in the renormalisation group analysis of critical trajectories for the above models below the critical dimension $d_c =2\alpha$. This revised version makes some changes, adds new material, and also corrects some errors in the previous version. It refers to the author's published article with the same title in J Stat Phys (2016) 163: 1235-1246, as well as to an erratum to be published in J Stat Phys.