Statistical mechanics, structure of matter Lattice spin systems Theorem

Aizenman–Higuchi theorem

Statement

Consider the Ising model on $\Z^2$ with nearest-neighbour coupling $J > 0$ and no external field, at inverse temperature $\beta$ . Let $\mu^+_\beta$ and $\mu^-_\beta$ be the infinite-volume Gibbs measures obtained as limits of finite boxes with all boundary spins $+1$ , respectively $-1$ . Then every Gibbs measure $\mu$ of the model at inverse temperature $\beta$ is a convex combination $\mu = \lambda \mu^+_\beta + (1 - \lambda) \mu^-_\beta \qquad \text{for some } \lambda \in [0, 1].$

Remarks

For $\beta \le \beta_c$ the two measures coincide and the Gibbs measure is unique, so the content is in the low-temperature phase $\beta > \beta_c$ . In particular, every Gibbs measure of the two-dimensional model is translation invariant. Michael Aizenman (1980) and Yasunari Higuchi (1981) proved the theorem independently. Both proofs use percolation of plus and minus spins and the topology of the plane, which together rule out a stable interface in two dimensions. The dimension is necessary. On $\Z^3$ , Dobrushin showed in 1972 that at low temperature there are Gibbs measures with a stable flat interface between a plus and a minus region. These measures are not translation invariant, so they are not of this form.

Going deeper

For a student

The result gives a complete description of the Gibbs measures of the two-dimensional Ising model, which is rare for a model with a phase transition. It contrasts with three dimensions, where Dobrushin states with rigid interfaces exist at low temperature. A later proof by Coquille and Velenik gives a shorter argument, and Georgii and Higuchi extended the percolation approach. The theorem is a good example of how planar topology helps in two-dimensional statistical mechanics.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`Z9EXV19CK2`
URI	`m2://Z9EXV19CK2`
Kind	statement, theorem
Writing	W0 stub
MSC	82B20, 82B26, 60K35
Status	This copy of the site has no build data. Run `m2 status Z9EXV19CK2`.
About	Ising model, Gibbs measure
Uses	Ising model, Gibbs measure
Changed	3 November 2026