 We then define a Hamiltonian by

Then, the random function with probability density proportional to formula_3 with respect to the Lebesgue measure on formula_4 is called the discrete GFF with boundary "U".It is not hard to show that the expected value formula_5 is the discrete harmonic extension of the boundary values from "U" (harmonic with respect to the transition kernel "P"), and the covariances formula_6 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".