In the one-dimensional zero-field <Ising model>, the bond variables $\tau_j=\sigma_j\sigma_{j+1}$ are independent and
$$
\mathbb E\tau_j=\frac{e^\beta-e^{-\beta}}{e^\beta+e^{-\beta}}=\tanh\beta.
$$
Since $\sigma_0\sigma_x=\prod_{j=0}^{x-1}\tau_j$ for $x>0$,
$$
\langle\sigma_0\sigma_x\rangle_{\beta,0}^{\mathbb Z}
=(\tanh\beta)^{|x|}
=e^{-c(\beta)|x|},
\qquad c(\beta)=-\log(\tanh\beta)>0.
$$

Solution

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

 Solution

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