Peierls argument

Statement

Consider the nearest-neighbor ferromagnetic Ising model on $\mathbb{Z}^d$ with $d\ge 2$, coupling $J>0$, and zero external field. Let $\mu_{\Lambda}^{+}$ denote the finite-volume Gibbs measure in a finite region $\Lambda$ with plus boundary condition.

Then there exists $\beta_0<\infty$ such that for all inverse temperatures $\beta>\beta_0$:

1. The probability (under plus boundary condition) that the origin is negative is small: $\mu_{\Lambda}^{+}(\sigma_0=-1)$ can be bounded by a convergent contour sum uniformly in large $\Lambda$.

2. Consequently, the infinite-volume plus state $\mu^{+}$ (any weak limit of $\mu_{\Lambda}^{+}$ as $\Lambda\uparrow\mathbb{Z}^d$) has strictly positive magnetization: $\mu^{+}(\sigma_0)>0$.

3. The plus and minus infinite-volume Gibbs states are distinct, so there are at least two infinite-volume Gibbs measures at low temperature: $\mu^{+}\neq \mu^{-}$.

In particular, the model exhibits phase coexistence / a phase transition for sufficiently large $\beta$.

Key hypotheses

• Dimension: $d\ge 2$ (so domain walls form nontrivial closed contours/surfaces).
• Ferromagnetic coupling: $J>0$ (aligning spins lowers energy).
• Zero field: $h=0$ (symmetry between $+$ and $-$ phases).
• Low temperature: $\beta$ large.

Key conclusions

• Peierls estimate: configurations with a “droplet” of $-$ spins inside $+$ boundary conditions pay an energy cost proportional to the size of the droplet boundary (a contour/surface).
• Spontaneous magnetization: $\mu^{+}(\sigma_0)>0$ (and similarly $\mu^{-}(\sigma_0)<0$).
• Non-uniqueness of Gibbs measures: there are multiple infinite-volume Gibbs states, typically extremal ones extremal Gibbs measures corresponding to $+$ and $-$ phases.

Proof idea / significance (sketch)

Work in a finite box $\Lambda$ with plus boundary condition. If $\sigma_0=-1$, then the cluster of $-$ spins containing the origin must be surrounded by an interface separating $-$ from $+$ spins. This interface can be encoded as a contour (in $d=2$) or a closed surface (in $d\ge 3$). Flipping all spins inside the contour changes the energy by at least a constant times the contour size, giving a weight ratio bounded by an exponential factor like $e^{-2\beta J|\Gamma|}$.

The remaining ingredient is counting: the number of distinct contours of size $n$ grows at most exponentially in $n$. Therefore the total probability that there exists a contour surrounding the origin is bounded by a geometric series $\sum_{n\ge n_0} (\text{const})^n e^{-2\beta J n}$, which converges for $\beta$ large. This implies $\mu_{\Lambda}^{+}(\sigma_0=-1)$ is small uniformly in $\Lambda$, forcing a positive limiting magnetization and hence coexistence of distinct infinite-volume phases.

Peierls’ argument is the classical mechanism behind low-temperature symmetry breaking in the Ising model.