This question is about a stochastic process theory. I really very bad in this topic. That's why I have to ask for help. I may mistranslate some terms but I'll do my best to give you right information.
The task is to find a limit of two-particle correlation function when $N\to\infty$ in a Gibbs model using the transfer-matrix method. That's all that I know. If I will help with some additional info, just ask me.
Upd 1. It seems that we need to find a limit of two-particle distribution function, since previous exercise is about one-particle corelation function which has no meaning.
I tried to solve this problem by myself, below are intermediate results of my search. Maybe it would help or at least someone will not say I didn't try.
I found that partition function is $Z(\beta) = tr(e^{-\sum_k\beta_k H_k})$, but don't understand how could matrix trace operator be applied to an expression inside the brackets, and don't know is it some general form or ready Gibbs model compliant one.
It seems that two-particle correlation function is $g_2(r_1,r_2,t) = n_2(r_1,r_2,t) - n_1(r_1,t)n_1(r_2,t)$ where
$$n_1(r,t) = \int{dX}\sum_{i=1}^N \delta(q_i - r)F(X,t)$$
$$n_2(r_1,r_2,t) = 1/2\int{dX}\sum_{i_1}\sum_{i_2\neq i_1} \delta(q_{i_1} - r_1)\delta(q_{i_2} - r_2)F(X,t)$$
Or maybe n1 and n2 are correlation functions itselves, I'm not sure. Formulas are from:
http://www.pd.isu.ru/sost/teor_phi/homepage/sinegovsky/lectures_tdsm/l15.pdf
(Doc is in Russian).
The more details you can provide the better, but I will really appreciate even an algorithm description. Just don't know in what direction to continue. Thanks.