Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 3 / iv / 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 functionIf the minimum pairwise separation tends to infinity, both displayed gaps tend to infinity. Since $\tanh \beta <1$, the expectation tends to zero.