Information from four point correlation functions in Ising model

Question

For a one-dimensional classical Ising model with the Hamiltonian $$H=-J \sum_{i}\sigma_{i} \, \sigma_{i+1}$$ where $\sigma=\left\{+1,-1\right\}$ one can calculate two point correlation for the spins $$\left<\sigma_{i} \, \sigma_{j}\right>.$$ I understand the meaning for this is that how two spins at different positions are correlated or in other words how fluctuations at the ${i}^{\text{th}}$ position affects the the spin at the position $j$.

Now, what is the physical meaning of four point correlation function $$\left<\sigma_{l} \, \sigma_{m} \, \sigma_{n} \, \sigma_{p}\right>.$$ What extra piece of information does it give? Can some explain intuitively?

Answer 1

Let me answer a more general question (which might not be what you are after...): what information is encoded in general correlation functions $\langle\sigma_A\rangle$, where $A$ is a finite set of vertices and $\sigma_A=\prod_{i\in A} \sigma_i$?

It turns out that one can prove (it's actually easy) that, for any local function $f$ (that is, any function depending only on finitely many spins), one can find (explicit) coefficients $(\hat{f}_A)_{A\subset\mathrm{supp}(f)}$ such that $$ f(\sigma) = \sum_{A\subset\mathrm{supp}(f)} \hat{f}_A \sigma_A $$ where $\mathrm{supp}(f)$ is the (finite) set of spins on which $f$ depends.

This means that knowing the correlation functions $\langle\sigma_A\rangle$ for every finite set $A$ allows you to compute the expectation of any local function $f$: $$ \langle f\rangle = \sum_{A\subset\mathrm{supp}(f)} \hat{f}_A \langle\sigma_A\rangle . $$ In this sense, the correlation functions $\langle\sigma_A\rangle$ contain all the information on the Gibbs measure.

(Let me emphasize that everything I said is completely general and not restricted to the one-dimensional model.)