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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.

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  • $\begingroup$ In "LQG AND Bosonic 2d string "( arxiv.org/pdf/1607.08467.pdf) , Vargas etal mention the construction of continuous GFF on compact manifolds. Then by projecting you obtain the DGFF. Now beyond compact, you lose the nice eigenfunction decomposition. $\endgroup$ – OOESCoupling Apr 5 '18 at 21:21
  • $\begingroup$ @ThomasKojar: I think this is the same idea as used in Sheffied's paper. My reservation is that I do not know how "by projecting..." works. If I put a triangulation on $M$, and I define GFF using Green functions via distributions, what does "projecting..." tell me? It is very unclear to me. $\endgroup$ – Bombyx mori Apr 6 '18 at 3:13
  • $\begingroup$ Projecting is made precise in their contour lines paper(arxiv.org/pdf/1008.2447.pdf). Basically we are projecting GFF on the subspace of H1 functions that are affine on triangles. This is how one shows convergence of DGFF to GFF. $\endgroup$ – OOESCoupling Apr 6 '18 at 20:58
  • $\begingroup$ @ThomasKojar: I spent sometime reading through his papers. However it is not entirely clear to me how the projecting procedure works even for 2D, where he used TG domains and did impose boundary conditions. Do you mind to turn your comment into an answer? $\endgroup$ – Bombyx mori Apr 7 '18 at 2:56
  • $\begingroup$ @ThomasKojar: My main confusion from reading his paper (arxiv.org/pdf/math/0605337.pdf) is that I do not see how the weights for the edges and vertices are being assigned, and how the DGFF relates to the metric from GFF. Do you mean that he just consider an element in GFF, then act it on a function in the subspace of $H^{1}$ functions that are affine on the triangles, and claim it is a way of defining DGFF? But then some hypothesis has to be put onto the triangles. $\endgroup$ – Bombyx mori Apr 7 '18 at 4:13
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Exactly what you want is here : https://arxiv.org/pdf/1809.03382.pdf

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    $\begingroup$ Would you mind expanding your answer? It is good if the answers here are not only link-only but in a way self contained. Thank you. $\endgroup$ – András Bátkai Oct 1 '18 at 14:57
  • $\begingroup$ It would be much better if you could make the answer more specific. $\endgroup$ – LeechLattice Oct 1 '18 at 16:27
  • $\begingroup$ I do hope there is some more expository work in the answer. But thanks! $\endgroup$ – Bombyx mori Oct 2 '18 at 2:28

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