I want to ask if a construction of discrete Gaussian free field has been done for a closed Riemannian manifold. Most of the literature I surveyed either need extra boundary condition and consider submanifold in $\mathbb{R}^{n}$, or consider discrete Gaussian free field on a "metrized graph" with weights on edges, faces and vertices. It is unclear to me, for example how the weight on the graphs related to the Riemannian metric on the underlying manifold. I am aware that there is no good discrete Laplacian that preserves all the properties of the connection Laplacian.

Since there are at least 9-10 different constructions of discrete Laplacian, I found it kind of odd that I could not find any work of discrete Gaussian free field on a closed manifold (there is a recent paper using bi-polar Green functions, which I found hard to digest).

With many thanks in advance.

I want to ask if a construction of discrete Gaussian free field has been done for a closed Riemannian manifold. Most of the literature I surveyed either need extra boundary condition and consider submanifold in $\mathbb{R}^{n}$, or consider discrete Gaussian free field on a "metrized graph" with weights on edges, faces and vertices. It is unclear to me, for example how the weight on the graphs related to the Riemannian metric on the underlying manifold. I am aware that there is no good discrete Laplacian that preserves all the properties of the connection Laplacian.

Since there are at least 9-10 different constructions of discrete Laplacian, I found it kind of odd that I could not find any work of discrete Gaussian free field on a closed manifold (there is a recent paper using bi-polar Green functions, which I found hard to digest).

With many thanks in advance.

Update: A quick "scan" showed that the new paper does not exactly do what I wanted (constructing a discrete Gaussian free field by choosing an appropriate grid that is compatible under refinement, the points chosen in this paper seems to be random). So the problem is still open.