Ferromagnetic monotonicity makes the plus-boundary expectation decrease as the finite volume grows, so
$$
\langle\sigma_0\rangle_{\beta,0}^{\mathbb Z^2,+}
=\lim_{L\to\infty}\langle\sigma_0\rangle_{\Lambda_L,\beta,0}^{+}
$$
exists. The uniform lower bound from part i passes to the limit, proving positive spontaneous magnetization at sufficiently large $\beta$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 4 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-209/4/a/ii/solution