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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-209/3/iv/solution