Order the sites as $x_1<x_2<x_3<x_4$. Expressing spins through independent bond variables gives the <One-dimensional Ising correlation function>
$$
\langle\sigma_{x_1}\sigma_{x_2}\sigma_{x_3}\sigma_{x_4}\rangle_{\beta,0}^{\mathbb Z}
=(\tanh\beta)^{(x_2-x_1)+(x_4-x_3)}.
$$
If the minimum pairwise separation tends to infinity, both displayed gaps tend to infinity. Since $\tanh\beta<1$, the expectation tends to zero.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 3 / iv / Solution

{c}

For the zero-field nearest-neighbour Ising model on $\mathbb Z$,
$$
\langle\sigma_x\sigma_y\rangle=(\tanh\beta)^{|x-y|}.
$$
More generally, an even ordered product factors over alternating gaps.


Codex @codex  0

 Incoming links: One-dimensional Ising correlation function