Gaussian Free Field

A Gaussian Free Field is a Gaussian Random Field as defined in Statistical Mechanics.

• AKA: GFF.
• See: Random Height Function, Gaussian Random Field, Markov Chain, Random Walk, Brownian Motion, Gaussian Process, Harmonic Function, Green's Function, Hamiltonian.

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References

2015

• (Wikipedia, 2015) ⇒ https://www.wikiwand.com/en/Gaussian_free_field
  • QUOTE: In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). Sheffield (2007) gives a mathematical survey of the Gaussian free field.

(...) Let P(x, y) be the transition kernel of the Markov chain given by a random walk on a finite graph G(V, E). Let U be a fixed non-empty subset of the vertices V, and take the set of all real-valued functions [math]\displaystyle{ \varphi }[/math] with some prescribed values on U. We then define a Hamiltonian by

[math]\displaystyle{ H( \varphi ) = \frac{1}{2} \sum_{(x,y)} P(x,y)\big(\varphi(x) - \varphi(y)\big)^2. }[/math]

Then, the random function with probability density proportional to [math]\displaystyle{ \exp(-H(\varphi)) }[/math] with respect to the Lebesgue measure on [math]\displaystyle{ \R^{V\setminus U} }[/math] is called the discrete GFF with boundary U.

It is not hard to show that the expected value [math]\displaystyle{ \mathbb{E}[\varphi(x)] }[/math] is the discrete harmonic extension of the boundary values from U (harmonic with respect to the transition kernel P), and the covariances [math]\displaystyle{ \mathrm{Cov}[\varphi(x),\varphi(y)] }[/math] are equal to the discrete Green's function G(x, y).

So, in one sentence, the discrete GFF is the Gaussian random field on V with covariance structure given by the Green's function associated to the transition kernel P.

2009

• (Macskassy, 2009) ⇒ Sofus A. Macskassy. (2009). “Using Graph-based Metrics with Empirical Risk Minimization to Speed Up Active Learning on Networked Data.” In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2009). doi:10.1145/1557019.1557087
  • QUOTE: Active and semi-supervised learning are important techniques when labeled data are scarce. Recently a method was suggested for combining active learning with a semi-supervised learning algorithm that uses Gaussian fields and harmonic functions.

2007

• (Sheffield, 2007) ⇒ Scott Sheffield. (2007) "Gaussian free fields for mathematicians." Probability theory and related fields 139.3-4 (2007): 521-541. [1]
  • QUOTE: The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm–Loewner evolution.

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