Peierls argument

Remarks

Under the plus boundary condition, if $\sigma_0 = -1$ then the origin is enclosed by a contour: a closed surface of dual plaquettes separating minus spins from plus spins. Flipping all spins inside a given contour $\gamma$ of size $\abs{\gamma}$ removes it and lowers the energy by $2J\abs{\gamma}$, so the probability that $\gamma$ is present is at most $e^{-2 \beta J \abs{\gamma}}$. The number of contours of size $\ell$ around the origin grows at most exponentially in $\ell$, say like $C^\ell$. So for $\beta$ large, $\mu^+_{\Lambda, \beta}(\sigma_0 = -1) \le \sum_\ell C^\ell e^{-2\beta J \ell}$, which is small uniformly in $\Lambda$. The argument uses $d \ge 2$: on $\Z$ a contour is just two points, and its energy cost does not grow with the enclosed region. Indeed the one-dimensional Ising model has a unique Gibbs measure at every temperature.

Going deeper

For a student

Rudolf Peierls gave the idea in 1936, and R. Griffiths and R. L. Dobrushin made it rigorous in 1964 and 1965. It is the prototype of contour methods and of Pirogov–Sinai theory for low-temperature phases. It shows that the critical inverse temperature $\beta_c$ of the Ising model is finite in dimension at least two. Combined with the high-temperature uniqueness from the Dobrushin criterion, it shows that $0 < \beta_c < \infty$.