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 $G_{\alpha } (x-y,m^2) = ((-\Delta )^{\alpha \over 2} + m^{2})^{-1} (x-y)$, 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.

Appendix

In this Appendix we prove the statements in the first paragraph of Remark 4. By definition the fluctuation covariances on the coarser scale $L'= L^{r}$ with $L\ge 2$ fixed and r a large positive integer is given by (1.17):

$$\begin{aligned} {\tilde{\Gamma }}'_{j,\alpha } (\cdot ,m^{2})= \sum _{l=0}^{r-1} {\tilde{\Gamma }}_{l+jr,\alpha } (\cdot , m^{2}). \end{aligned}$$

(3.20)

Therefore we get

$$\begin{aligned} \Big \Vert {\partial \over \partial m^{2}} \partial _{ {{\mathbb Z}}^{d} }^{p} {\tilde{\Gamma }}'_{j,\alpha }(\cdot , m^{2}) \Big \Vert _{L^{\infty } ({{\mathbb Z}}^{d})}\le & {} \sum _{l=0}^{r-1} \Big \Vert {\partial \over \partial m^{2}} \partial _{ {{\mathbb Z}}^{d} }^{p} {\tilde{\Gamma }}_{l+jr,\alpha } (\cdot , m^{2}) \Big \Vert _{L^{\infty } ({{\mathbb Z}}^{d})} \\\le & {} c_{L, \alpha , p} (m^{2})^{-2(1-{1\over \alpha })} \sum _{l=0}^{r-1} L^{-p(l+jr)} L^{-(l+jr)(d-2)} \nonumber \\\le & {} c_{L, \alpha , p} (m^{2})^{-2(1-{1\over \alpha })} (L')^{-pj} (L')^{(d-2)j} \sum _{l=0}^{\infty } L^{-(p +(d-2))l}. \end{aligned}$$

For $d\ge 3$, $\forall p \ge 0$ and $d=2$, $\forall p \ge 1$ we can bound the sum on the right hand side by

$$\begin{aligned} \sum _{l=0}^{\infty } L^{-l} = \Big (1-{1\over L}\Big )^{-1} \end{aligned}$$

and hence

$$\begin{aligned} \Big \Vert {\partial \over \partial m^{2}} \partial _{ {{\mathbb Z}}^{d} }^{p} {\tilde{\Gamma }}'_{j,\alpha }(\cdot , m^{2}) \Big \Vert _{L^{\infty } ({{\mathbb Z}}^{d})} \le c'_{L, \alpha , p} (m^{2})^{-2(1-{1\over \alpha })} (L')^{-pj} (L')^{(d-2)j} . \end{aligned}$$

which is (1.20) with a new constant independent of $L'$ as claimed. $\square$