David C. Brydges, G. Guadagni, P.K. Mitter
Finite Range Decomposition of Gaussian Processes
(81K, LaTeX2e)
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 $n\rightarrow
\infty$. 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.