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

In the one-dimensional zero-field Ising model, the bond variables ${\tau }_j={\sigma }_j{\sigma }_{j+1}$ are independent andSince ${\sigma }_0{\sigma }_x={\prod }_{j=0}^{x-1}{\tau }_j$ for $x>0$,

In a finite interval, the effect at the origin of a fixed boundary spin a distance $N$ away is bounded by its two-point correlation, $(\tanh \beta {)}^N$. With two boundaries,Thus the plus and minus boundary limits coincide with the unique spin-flip-symmetric one-dimensional Gibbs state, and $⟨{\sigma }_0{⟩}_{\beta ,0}^{\mathbb{Z},\pm }=0$ for every finite $\beta$.

The product ${\sigma }_0{\sigma }_1\cdots {\sigma }_{2022}$ contains $2023$ spins and changes sign under the global spin flip $\sigma \mapsto -\sigma$. By part ii, the infinite-volume plus state is spin-flip symmetric. Therefore

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.