Discrete approximation of Minkshisundaram-Pleijel zeta function? I'm looking for some references on the following situation: 
$S$ is a Riemannian surface, and $G_n$ is a sequence of metric subgraphs embedded on $S$. Let $\zeta_n$ be the zeta function of the Laplacian of $G_n$ (Mellin transform of $Tr (h_t) - 1$, where $h_t$ is the heat kernel of $G_n$), and let $\zeta_S$ be the Minakshisundaram–Pleijel zeta function of $S$.
Under what conditions and for what modes of convergence do we have $\zeta_n(s)  \to \zeta(s)$, $\zeta_n'(s) \to \zeta'(s)$, etc. 
I'm particularly interested in the convergence of $exp( - \zeta_n'(0) ) / |G_n| \to exp( - \zeta'(0)) / Vol(S)$. 
(This relates to my question here: Does the zeta regularized Laplacian determinant measure the volume of some parameter space? How many "spanning trees" on a manifold? )
 A: This is an old idea that has been investigated thoroughly by Patodi and  Dodziuk. 
See their paper for more details. But what they proved is more subtle (about Hodge Laplacian vs metric Laplacian). Later Muller used their techniques to prove his part of the Cheeger-Muller theorem. 
For the volume ratio analogy I think I answered in the previous post. The paper I mentioned there specifically showed that minus some exceptions, we should have (2.6.1)
$$
\frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi}
$$
if their conjecture holds based on another result of Muller. But I think some (non-artihemetic) counter-examples have been found already. 
A: I learned of a few results that give useful answers to (the spirit of) this question:
Zeta functions, heat kernels and spectral asymptotics on
degenerating families of discrete tori ( G. Chinta, J. Jorgenson, and A. Karlsson ) : https://arxiv.org/pdf/0806.2014.pdf
The asymptotic determinant of the discrete Laplacian (Richard Kenyon):
https://arxiv.org/abs/math-ph/0011042
Generally these results look like the following:
We consider a natural sequence of graph $G_n$ converging to a surface $U$. The asymptotics of the log number of spanning trees of $G_n$ is given (roughly) by $[\text{some geometric properties of U}] n^2  + \text{laplacian determinant of U} + constants + o(1)$. 
(For the rectangular grid graph case, see especially proposition 13 in Kenyon's paper and compare to formula 105 here: https://projecteuclid.org/download/pdf_1/euclid.cmp/1104271130 ]
The papers I linked treat the case where $U$ is a torus or a rectilinear polygon in $\mathbb{R}^2$, and $G_n$ is either a sequence of discrete tori, or subgraph $\epsilon \mathbb{Z}^2 \cap U$.
I'd be really interested in seeing to what extent these can be generalized...
