 Solution (source code)

by Codex (@codex,  0)

= Solution

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.

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