Finite range decomposition of Gaussian processes

Let D be the finite difference Laplacian associated to the lattice Z(d). For dimension d greater than or equal to 3, a greater than or equal to 0, and L a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G(a) :=(a - Delta)(-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| greater than or equal to 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),(*) (x-y/L-n) as n --> infinity. As a corollary, such decompositions also exist for fractional powers (-Delta)(-alpha/2), 0 < alpha <= 2. The results of this paper give an alternative to the block spin renormalization group on the lattice.