Finite Range Decomposition of Gaussian Processes

Abstract

Let ∆ 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 − ∆) −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 formx−y L n as n → ∞. As a corollary, such decompositions also exist for fractional powers (The results of this paper give an alternative to the block spin renormalization group on the lattice.

Erratum to: On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian

“…The coarse scale L independence with respect to the bounds in [3] was found earlier in R. Bauerschmidt (unpublished data) for d ≥ 3 but in d = 2 an additional log L dependence was found.…”

“…The references to equations are to those in the above paper. Reference [3] cited in the above paper is cited as Ref. [3] in this erratum.…”

“…Reference [3] cited in the above paper is cited as Ref. [3] in this erratum. The statement of Theorem 1.1 in [1] has some modifications as a result of these corrections.…”

“…We also take this opportunity to correct errors about constants. We first state them: If we follow the decomposition in [3], instead of [4], and this will be the case here, then the constants c p which occur in Sect. 3 should read c L , p .…”

“…In Appendix A of [3] interior regularity estimates (like those of Nirenberg and Agmon in the continuum) were obtained for the solution of a lattice Dirichlet problem for the minus lattice laplacian plus a mass squared parameter (called a ≥ 0). This is called s in the present paper.…”