1206.2212 (Roland Bauerschmidt)

A simple method for finite range decomposition of quadratic forms and
Gaussian fields    [PDF]

Roland Bauerschmidt

The Gaussian free field associated to a symmetric Dirichlet form is a Gaussian process whose covariance is the quadratic form dual to the Dirichlet form, which we refer to as the corresponding Green's form. The free field is of significance for many models of statistical mechanics. We present a simple method to decompose the Green's forms of a large class of interesting symmetric Dirichlet forms into integrals over symmetric, non-negative definite, and finite range (properly supported) forms that are smoother than the original form. Our method makes use of the finite propagation speed of the wave equation and Chebyshev polynomials. It yields concise proofs of results of [4,9,7,1], but gives effective estimates more generally. This result gives rise to multiscale decompositions in distribution of the associated Gaussian free fields into sums of independent smoother Gaussian fields with spatially localized correlations.

View original: http://arxiv.org/abs/1206.2212