Discrete Gaussian Free Field via Hadamard's formula

TL;DR

This paper introduces a new method to construct the Discrete Gaussian Free Field on weighted graphs using a dynamical Green function expansion, leading to a dimension-free Hadamard variation formula applicable to fractal geometries.

Contribution

It presents a novel, dimension-free discrete Hadamard variational formula and a construction of the Gaussian Free Field that does not require smoothness or specific dimension constraints.

Findings

Discrete Hadamard variation formula derived

Dimension-free construction of Gaussian Free Field

Potential extension to fractal geometries

Abstract

We present a novel way of constructing the Gaussian Free Field on a weighted graph via a dynamical expansion of the Green function along an expanding family of subgraphs. Along the way we obtain the discrete analogue of the classical Hadamard variational formula regarding the variation of the Green function under infinitesimal variations of the domain. In order to develop necessary machinery we construct expanding bases of the naturally associated energy spaces. An interesting observation is that both our discrete Hadamard variation formula and and the related construction of the discrete Gaussian Free Field are completely dimension-free and do not require smoothness of any kind. The graph model contains geometric information via the edges which supply the discrete topological information, and by conductances which give metric information. Going to a continuum limit, we would then obtain continuous version of the Hadamard variational formula and the associated Hadamard operator in e.g. fractal geometries of arbitrary dimension.