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

In previous papers, Mitter (J Stat Phys 163:1235–1246, 2016; Erratum: J Stat Phys 166:453–455, 2017; On a finite range decomposition of the resolvent of a fractional power of the Laplacian, http://arxiv.org/abs/1512.02877), we proved 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 \({{\mathbb Z}}^{d}\) 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 \({{\mathbb Z}}^{d}/L^{N+1}{{\mathbb Z}}^{d} \) 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.