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

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 $\beta$.