Under <plus boundary condition>[plus boundary conditions], if $\sigma_0=-1$, the negative cluster containing the origin is surrounded by a <Peierls contour> $\gamma$. Flipping all spins inside $\gamma$ is injective after $\gamma$ is specified and increases the Boltzmann weight by $e^{2\beta|\gamma|}$. Therefore
$$
\mathbb P_{\Lambda_L,\beta}^{+}(\sigma_0=-1)
\leq\sum_{\gamma\ni0}e^{-2\beta|\gamma|}.
$$
The number of length-$m$ contours surrounding the origin is at most $C m3^m$, so the sum tends to zero as $\beta\to\infty$, uniformly in $L$. For sufficiently large $\beta$ it is below $1/4$, and then
$$
\langle\sigma_0\rangle_{\Lambda_L,\beta}^{+}
=1-2\mathbb P(\sigma_0=-1)\geq\frac12.
$$

Solution

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

 Solution

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