Gaussian free field
In probability theory and statistical mechanics, the Gaussian free field is a Gaussian random field, a central model of random surfaces.
The discrete version can be defined on any graph, usually a lattice in d-dimensional Euclidean space. The continuum version is defined on Rd or on a bounded subdomain of Rd. It can be thought of as a natural generalization of one-dimensional Brownian motion to d time dimensions: it is a random function from Rd to R. In particular, the one-dimensional continuum GFF is just the standard one-dimensional Brownian motion or Brownian bridge on an interval.
In the theory of random surfaces, it is also called the harmonic crystal. It is also the starting point for many constructions in quantum field theory, where it is called the Euclidean bosonic massless free field. A key property of the 2-dimensional GFF is conformal invariance, which relates it in several ways to the Schramm–Loewner evolution, see and.
Similarly to Brownian motion, which is the scaling limit of a wide range of discrete random walk models, the continuum GFF is the scaling limit of not only the discrete GFF on lattices, but of many random height function models, such as the height function of distribution (discrete)|uniform random] planar domino tilings, see. The planar GFF is also the limit of the fluctuations of the characteristic polynomial of a random matrix model, the Ginibre ensemble, see.
The structure of the discrete GFF on any graph is closely related to the behaviour of the simple random walk on the graph. For instance, the discrete GFF plays a key role in the proof by of several conjectures about the cover time of graphs.
Definition of the discrete GFF
Let P be the transition kernel of the Markov chain given by a random walk on a finite graph G. Let U be a fixed non-empty subset of the vertices V, and take the set of all real-valued functions with some prescribed values on U. We then define a Hamiltonian byThen, the random function with probability density proportional to with respect to the Lebesgue measure on is called the discrete GFF with boundary U.
It is not hard to show that the expected value is the discrete harmonic extension of the boundary values from U, and the covariances are equal to the discrete Green's function G.
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.
The continuum field
The definition of the continuum field necessarily uses some abstract machinery, since it does not exist as a random height function. Instead, it is a random generalized function, or in other words, a probability distribution on distributions.Given a domain Ω⊆Rn, consider the Dirichlet inner product
for smooth functions ƒ and g on Ω, coinciding with some prescribed boundary function on, where is the gradient vector at. Then take the Hilbert space closure with respect to this inner product, this is the Sobolev space.
The continuum GFF on is a Gaussian random field indexed by, i.e., a collection of Gaussian random variables, one for each, denoted by, such that the covariance structure is for all.
Such a random field indeed exists, and its distribution is unique. Given any orthonormal basis of, we can form the formal infinite sum
where the are i.i.d. standard normal variables. This random sum almost surely will not exist as an element of, since if it did then
However, it exists as a random generalized function, since for any we have
hence
is a centered Gaussian random variable with finite variance
Special case: ''n'' = 1
Although the above argument shows that does not exist as a random element of, it still could be that it is a random function on in some larger function space. In fact, in dimension, an orthonormal basis of is given byand then is easily seen to be a one-dimensional Brownian motion. So, in this case, it is a random continuous function. For instance, if is the Haar basis, then this is Lévy's construction of Brownian motion, see, e.g., Section 3 of.
On the other hand, for it can indeed be shown to exist only as a generalized function, see.