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

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