PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT

Percolation and Disordered Systems 259
12. ISING AND POTTS MODELS
12.1 Ising Model for Ferromagnets
In a famous experiment, a piece of iron is exposed to a magnetic field. The
field increases from zero to a maximum, and then diminishes to zero. If the
temperature is sufficiently low, the iron retains some ‘residual magnetisation’,
otherwise it does not. The critical temperature for this phenomenon is often
called the Curie point. In a famous scientific paper [195], Ising proposed a
mathematical model which may be phrased in the following way, using the
modern idiom.
Let Λ be a box of Z d , say Λ = [−n, n] d . Each vertex in Λ is allocated a random
spin according to a Gibbsian probability measure as follows. Since spins
come in two basic types, we take as sample space the set ΣΛ = {−1, +1} Λ ,
and we consider the probability measure πΛ which allocates a probability to
a spin vector σ ∈ ΣΛ given by
(12.1) πΛ(σ) = 1
ZΛ
exp{−βHΛ(σ)}, σ ∈ ΣΛ,
where β = T −1 (the reciprocal of temperature, on a certain scale) and the
Hamiltonian HΛ : ΣΛ → R is given by
(12.2) HΛ(σ) = − �
e=〈i,j〉
Jeσiσj − h �
for constants (Je) and h (called the ‘external field’) which parameterise the
process. The sums in (12.2) are over all edges and vertices of Λ, respectively.
The measure (12.1) is said to arise from ‘free boundary conditions’, since
the boundary spins have no special role. It turns out to be interesting to allow
other types of boundary conditions. For any assignment γ : ∂Λ → {−1, +1}
there is a corresponding probability measure π γ
Λ obtained by restricting the
vector σ to the set of vectors which agree with γ on ∂Λ. In this way we
may obtain measures π +
Λ , π−
Λ , and πf Λ (with free boundary conditions) on
appropriate subsets of ΣΛ.
For simplicity, we assume here that Je = J > 0 for all edges e. In this
‘ferromagnetic’ case, measures of the form (12.1) prefer to see configurations
σ in which neighbouring vertices have like spins. The antiferromagnetic case
J < 0 can be somewhat tricky.
Inspecting (12.1)–(12.2) with J > 0, we see that spins tend to align with
the sign of any external field h.
The following questions are basic.
(a) What weak limits limΛ→Zd π γ
Λ exist for possible boundary conditions
γ? (This requires redefining π γ
Λ as a probability measure associated
with the sample space Σ = {−1, +1} Zd.)
i
σi