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A characterisation of the continuum Gaussian free field in $d \geq 2$ dimensions

Juhan Aru, Ellen Powell

Abstract

We prove that under certain mild moment and continuity assumptions, the $d$-dimensional Gaussian free field is the only stochastic process in $d\geq 2$ that is translation invariant, exhibits a certain scaling, and satisfies the usual domain Markov property. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

A characterisation of the continuum Gaussian free field in $d \geq 2$ dimensions

Abstract

We prove that under certain mild moment and continuity assumptions, the -dimensional Gaussian free field is the only stochastic process in that is translation invariant, exhibits a certain scaling, and satisfies the usual domain Markov property. Our proof is based on a decomposition of the underlying functional space in terms of radial processes and spherical harmonics.

Paper Structure

This paper contains 12 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Immediate consequences
  3. The domain Markov decomposition is unique.
  4. The 2-point function.
  5. The 4-point function.
  6. The diagonal contribution of the 2-point function is negligible.
  7. Spherical averages.
  8. Covariance is the Green function
  9. Gaussianity
  10. Weak Gaussianity in terms of the 4-point function
  11. Gaussianity of spherical averages
  12. Gaussianity in the general case

Key Result

Theorem \oldthetheorem

Let $d \geq 2$ and $\mathbb{B} \subseteq \mathbb{R}^d$ denote the open unit ball. Suppose that $h$ is a random Schwartz distribution with support on $\mathbb{B}$, i.e. a random continuous functional $(h,f)$ indexed by $f \in C_c^\infty(\mathbb{R}^d)$ and zero as soon as $f$ has support oustide of $\

Theorems & Definitions (26)

  • Definition 1: Gaussian free field
  • Definition \oldthetheorem: Domain Markov Property (DMP) with scaling for balls
  • Theorem \oldthetheorem: Characterisation of the $d$-dimensional GFF
  • Remark \oldthetheorem
  • Definition \oldthetheorem: Scaling function
  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • proof : Proof of \ref{['4mom_av']}
  • ...and 16 more