The discrete Gaussian free field on a compact manifold

TL;DR

This paper extends the concept of the discrete Gaussian free field to compact manifolds by constructing a suitable random graph approximation and proves convergence to the continuum Gaussian free field in the scaling limit.

Contribution

It introduces a method to define the DGFF on manifolds using random graphs and demonstrates convergence to the continuum GFF in Sobolev space topology.

Findings

Construction of a random graph approximation for manifolds

Proof of convergence of DGFF to the continuum GFF

Interpretation of DGFF as an element of Sobolev space

Abstract

In this article we aim at defining the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice $\mathbb{Z}^d$ in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.