Finite range Decomposition of Gaussian Processes

Abstract

Let $\text{D}$ be the finite difference Laplacian associated to the lattice ${\text{bZ}}^d$. For dimension $d\ge 3$, $a\ge 0$ and $L$ a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent $G^a:=(a-\text{D}{)}^{-1}$ can be decomposed as an infinite sum of positive semi-definite functions $V_n$ of finite range, $V_n(x-y)=0$ for $|x-y|\ge O(L{)}^n$. Equivalently, the Gaussian process on the lattice with covariance $G^a$ admits a decomposition into independent Gaussian processes with finite range covariances. For $a=0$, $V_n$ has a limiting scaling form $L^{-n(d-2)}{\Gamma }_{c,\ast }\left(\frac{x-y}{L^n}\right)$ as $n\to \infty$. As a corollary, such decompositions also exist for fractional powers $(-\text{D}{)}^{-\alpha /2}$, $0<\alpha \le 2$. The results of this paper give an alternative to the block spin renormalization group on the lattice.

Comments: 26 pages, LaTeX, paper in honour of G.Jona-Lasinio.Typos corrected, corrections in section 5 and appendix A