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Wick theorem for analytic functions of Gaussian fields

Fabio Coppini, Wioletta M. Ruszel, Dirk Schuricht

Abstract

We compute the correlation of analytic functions of general Gaussian fields in terms of multigraphs and Feynman diagrams on the lattice Z^d. Then, we connect its scaling limit to tensors of the correlation functionals of Fock space fields. Afterwards, we investigate the relation with fermionic Gaussian field states for even functions. For instance, we characterize the correlation functionals of the exponential of a continuous Gaussian Free Field or general analytic functions of fractional Gaussian fields as limits of quantities constructed via a sequences of discrete fields. Finally, we show that the duality between even powers of bosonic Gaussian fields and "complex" fermionic Gaussian fields can be reformulated in terms of a principal minors assignment problem of the corresponding covariance matrices.

Wick theorem for analytic functions of Gaussian fields

Abstract

We compute the correlation of analytic functions of general Gaussian fields in terms of multigraphs and Feynman diagrams on the lattice Z^d. Then, we connect its scaling limit to tensors of the correlation functionals of Fock space fields. Afterwards, we investigate the relation with fermionic Gaussian field states for even functions. For instance, we characterize the correlation functionals of the exponential of a continuous Gaussian Free Field or general analytic functions of fractional Gaussian fields as limits of quantities constructed via a sequences of discrete fields. Finally, we show that the duality between even powers of bosonic Gaussian fields and "complex" fermionic Gaussian fields can be reformulated in terms of a principal minors assignment problem of the corresponding covariance matrices.

Paper Structure

This paper contains 25 sections, 16 theorems, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Aim of this work and related literature
  3. Informal statements of the main results
  4. Organization of the paper
  5. Wick theorem for analytic functions of Gaussian fields
  6. The Wick product
  7. Feynman diagrams and multigraphs
  8. Main result
  9. Example: the discrete Gaussian free field
  10. Example: the discrete fractional Gaussian field
  11. On the properties of the cumulants
  12. Scaling limit of the $k$-point function
  13. Example: discrete Gaussian Free Field and its gradient
  14. Example: discrete membrane model
  15. Example: discrete fractional Gaussian fields
  16. ...and 10 more sections

Key Result

Theorem 1

For a general Gaussian field with covariance matrix $G$ defined on $\Lambda \subset \mathbb{Z}^d$, $d\;\geqslant\; 2$, and $f(x) = \sum_{n} a_n x^n$, we have that for $N$ points $x_1, \dots, x_N$ in $\Lambda$, it holds where $c_{f,N}=\prod_{i=1}^{N} a_{n_i}$ and $c_q=const(q)$ is an explicit constant. $q$ is an undirected multigraph with no self-loops.

Figures (2)

  • Figure 1: Two different Feynman diagrams, labeled by the centered Gaussian random variables $X_1, \dots, X_4$. Although different, $\gamma_1$ and $\gamma_2$ have the same value, i.e, $\nu(\gamma_1) = \nu(\gamma_2) = \prod_{1\;\leqslant\; i<j\;\leqslant\; 4} {\mathbf E} \left[X_i X_j\right]^2$.
  • Figure 2: Examples of two distinct connected multigraphs on 4 vertices and 4 edges per vertex. Each vertex has $4$ nodes that shall not be connected with each others, i.e., no self-loop. These nodes are exchangeable at the level of the graph, i.e., if you rewire an edge from one node to another in the same vertex, the multigraph remains the same. However, they do play a role in the computation of Gaussian correlation functions as exchanging them yield a different Feynman diagram. See the proof of Theorem \ref{['thm:wick-analytic']} for more on that.

Theorems & Definitions (28)

  • Theorem : see Theorem \ref{['thm:wick-analytic']}
  • Proposition : see Propositions \ref{['pro:non-triv-cumulants']} and \ref{['pro:non-triv-cumulants-fock']}
  • Theorem : see Theorem \ref{['thm:cumulants-p=2']}
  • Lemma : see Lemma \ref{['lem:bos-fer-p>2']}
  • Theorem 2.1: janson_gaussian_1997
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 2.4
  • proof
  • ...and 18 more