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

Abstract

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 Gα(x-y,m2)=((-Δ)α2+m2)-1(x-y), for 0 < α< 2 and all real m, in the lattice Zd for dimension d≥ 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 Zd/ LN+1Zd for d≥ 2 provided m≠ 0 and 0 < α< 2. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter m2. Here L is any odd positive integer and N≥ 2 is any positive integer.