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

Under 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 $e^{2\beta |\gamma |}$. ThereforeThe number of length-$m$ contours surrounding the origin is at most $Cm{3}^m$, so the sum tends to zero as $\beta \to \infty$, uniformly in $L$. For sufficiently large $\beta$ it is below $1/4$, and then

Ferromagnetic monotonicity makes the plus-boundary expectation decrease as the finite volume grows, soexists. The uniform lower bound from part i passes to the limit, proving positive spontaneous magnetization at sufficiently large $\beta$.

The three-dimensional Peierls argument replaces planar contours by closed dual plaquette surfaces surrounding the negative cluster of the origin. A surface of area $m$ costs $e^{-2\beta m}$, and the number of connected surfaces of area $m$ containing a fixed nearby plaquette is at most $C^m$. HenceThis is below $1/2$ for sufficiently large $\beta$, so $⟨{\sigma }_0{⟩}_{\beta ,0}^{{\mathbb{Z}}^3,+}>0$.