Finite Range Decomposition of Gaussian Processes

Abstract

Let Δ be the finite difference Laplacian associated to the lattice Z ^d. For dimension d≥3, a≥0, and L a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G ^a≔(a−Δ)⁻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|≥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 $L^{ - n\left( {d - 2} \right)} \Gamma _{c,*} \left( {\tfrac{{x - y}}{{L^n }}} \right)$ as n→∞. As a corollary, such decompositions also exist for fractional powers (−Δ)^−α/2, 0<α≤2. The results of this paper give an alternative to the block spin renormalization group on the lattice.