\[sec:dfgf\], we construct a discrete version of $\operatorname{FGF}_s$ for $s\in
(0,1)$ by introducing a discrete fractional gradient to play the role of the discrete gradient in the definition of the discrete GFF. The fractional gradient involves long range interactions, which may be viewed as the reason that the Markov property fails for $\operatorname{FGF}_s$ when $s$ is not an integer.

The comparison between the short range $\operatorname{FGF}_s({\mathbb{R}}^d)$ (when $s\in {\mathbb{Z}}$) and the long range $\operatorname{FGF}_s({\mathbb{R}}^d)$ (when $s\notin {\mathbb{Z}}$) may also be seen from the point of view of the corresponding potential theories. As an illustration, consider GFF and $\operatorname{FGF}_s$ for $0<s<1$. The covariance kernel for the Gaussian free field is given by the solution of the ordinary Laplace equation $-\Delta f = \phi$. As we will see, the counterpart for $\operatorname{FGF}_s$ with $0<s<1$ is the fractional Laplacian equation $(-\Delta)^{s} f=\phi$. The Laplacian is a local differential operator, while $(-\Delta)^{s}$ for $s\in (0,1)$ is a non-local pseudo-differential operator and $(-\Delta)^s
f(x)$ depends on the values of $f(x)$ for all $x\in {\mathbb{R}}^d$. Another way to see the distinction between the $s=1$ and $s\in(0,1)$ cases is to recall that the Green’s function for the Dirichlet Laplacian is given by the density of the occupation 