Gaussian Free Field - Definition of The Discrete GFF

Definition of The Discrete GFF

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 φ with some prescribed values on U. We then define a Hamiltonian by

Then, the random function with probability density proportional to exp(−H(φ)) with respect to the Lebesgue measure on RV−U 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 (harmonic with respect to the transition kernel P), and the covariances Cov 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.