2 added sparse explaination and introduction

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, GiardinĂ , van der Hofstad, 2010) the partition function of the Ising model on a sparse random graph was calculated...wellcalculated (sparse = almost surely no triangle, quadrangle etc )...well, I must add that "calculated" refers to a random-variable fix-point fixpoint equation - I've put some effort into deriving an explicit expression thereof, but did not suceed so far, but that's ... 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?

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# Correlation-Function for Random Graph Ising Model

In this paper (Dommers, GiardinĂ , van der Hofstad, 2010) the partition function of the Ising model on a random graph was calculated...well, I must add that "calculated" refers to a random-variable fix-point equation - I've put some effort into deriving an explicit expression thereof, but did not suceed so far, but that's 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?