For non-Ising'ers: Given a graph, we study the probability-distribution on the set of colorings ("Spin-up" and "-down") generated by a given correlation ("force to equality") between adjacient nodes (Bolzmann-Term depending exponentially on a "Temperature Parameter"). The limit of high Temperature is no-correlation whatsoever, the limit of low Temerature is all colors coincide for sure (are "parallel").
In this paper (Dommers, Giardinà, van der Hofstad, 2010) the partition function of the Ising model on a sparse random graph was calculated (sparse = almost surely no triangle, quadrangle etc )...well, I must add "calculated" refers to a random-variable fixpoint equation - I've put some effort into deriving an explicit expression thereof, but did not suceed ... a different story and maybe later a different question ;-)
The proof techniques very basically relay on the transfer matrix method for a local tree inside the random graph (which is defined by a given branching distribution, e.g. Poisson); this ultimately leads to a recursion formula for the up/down-random-variable of a single site and the authors proof the existence of a unique solution (the mentionen fix-point). Partition function etc. are derived as various expectation vals.
For use in an own work I would like to get ahold of the correlators, i.e. the probability of two sites of given distance $a$ to coincide. Though I do not expect this to be easier than explicitely calculate the partition function, a similar fix-point equation as in the original work would totally suffice! I tried the trick one uses in 1D, but I didn't get through...although I would naively expect, that it's easier than 2D, as you have better control over the walks...?
Moreover, it would be interesting to have multi-correlators in the sense of knowing the probability of given up-down configurations e.g. in an arbitrary $abc$-triangle depending on the edge-length's $a,b,c$. $\;\;$ I don't even recall such from classical (2D) Ising model....any sources or ideas? Is that in "physically-sound" language the n-point-functions in this case?