Differentiate the finite-volume magnetization:
$$
\frac d{dh}\langle\sigma_x^1\rangle_{\Lambda,\beta,h}
=\sum_{y\in V}
\left\{
\langle\cos\theta_x\cos\theta_y\rangle
-\langle\cos\theta_x\rangle\langle\cos\theta_y\rangle
\right\}.
$$
Every summand is nonnegative by the <Ginibre inequality> with $a=\mathbf1_x$ and $b=\mathbf1_y$. Hence the magnetization is nondecreasing for $h\geq0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 209 / 1 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-209/1/ii/solution