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Discrete Gaussian Free Field via Hadamard's formula

Haakan Hedenmalm, Pavel Mozolyako, Daniil Panov

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.

Discrete Gaussian Free Field via Hadamard's formula

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.
Paper Structure (27 sections, 10 theorems, 85 equations)

This paper contains 27 sections, 10 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. The DGFF process
  3. Growing the DGFF along a discrete foliation
  4. Structure of the paper
  5. Discrete setting and notation
  6. Purpose of this section
  7. Graphs with conductances
  8. Hilbert spaces on vertices and on edges
  9. Weighted boundary and coboundary operators
  10. The discrete Laplacian
  11. The discrete Green function
  12. The discrete Poisson kernel
  13. Foliating the graph $\Gamma$
  14. Simplification of the notational conventions for discrete foliations
  15. Orthonormal bases in $H^1_0$: construction of the foliation
  16. ...and 12 more sections

Key Result

Proposition 2.1

Let $f\in \ell^2(U),\phi\in \ell^2_{-}(E_U)$. Then: Moreover, if $g$ is any function on $V$, then

Theorems & Definitions (16)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • Proposition 4.1
  • proof
  • ...and 6 more