Finite range Decomposition of Gaussian Processes

TL;DR

This paper proves that the resolvent of the lattice Laplacian in dimensions three and higher can be decomposed into an infinite sum of finite-range positive semi-definite functions, providing an alternative to block spin renormalization.

Contribution

It introduces a novel finite range decomposition of the Gaussian process covariance associated with the lattice Laplacian, applicable to fractional powers and offering a new approach to renormalization.

Findings

Decomposition of the resolvent into finite-range functions for $d \\ge 3$

Existence of a limiting scaling form for the decomposition components

Applicable to fractional powers of the Laplacian, including $a=0$ case

Abstract

Let $\D$ be the finite difference Laplacian associated to the lattice $\bZ^{d}$. 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 $G^{a}:=(a-\D)^{-1}$ 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 $L^{-n(d-2)}\Gamma_{c,\ast}{\bigl (\frac{x-y}{L^{n}}\bigr)}$ as $n\to \infty$. As a corollary, such decompositions also exist for fractional powers $(-\D)^{-\alpha/2}$, $0<\alpha \leq 2$. The results of this paper give an alternative to the block spin renormalization group on the lattice.