Gaussian Free Field - The Continuum Field

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 distribution on distributions (with two different meanings of the word "distribution").

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 (with the given boundary condition), 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 its variance is infinite. However, it exists as a random generalized function, since for any we have

hence

is a well-defined finite random number.