A Course on Large Deviations with an Introduction to Gibbs Measures.

4.3. Curie-Weiss model for ferromagnetism 37
Exercise 4.13. Let ηk > 0 be any sequence converging to 0, and let ρk be
a probability measure supported on the interval [−ηk, ηk]. Let {Xk} be a
sequence of independent random variables such that Xk has distribution ρk.
Let Sn be the partial sum as before and µn the distribution of Sn/n. Show
LDP(µn, n, I) holds with rate function I(0) = 0, I(x) = ∞ for x = 0.
We generalize this in Chapter 15. For example, the above result will continue
to hold under weak convergence ρk → δ0, as long as these distributions
have a common compact support.
But show that if Xk has distribution P {Xk = 0} = 1 − ηk and P {Xk =
k} = ηk, then the above rate function does not work if ηk converges to zero
slowly enough.
4.3. Curie-Weiss model for ferromagnetism
In this section we look at a simple model of spontaneous magnetization to
illustrate the usefulness of Cramér’s theorem and Varadhan’s theorem in
studying models from statistical mechanics.
For each n ∈ N, we have a model of n atoms, j = 1, . . . , n, each of which
has a spin ωi ∈ {−1, 1}. The space of spin configurations is denoted by
Ωn = {−1, 1} n . The energy of the system is given by the Hamiltonian
Hn(ω) = − J
2n
1≤i,j≤n
ωiωj − h
n
ωj,
where J > 0 and h ∈ R are constants. (J > 0 corresponds to ferromagnetism
and h is the strength of the external magnetic field.)
Nature prefers lower energy states, so in a ferromagnetic material the
spins tend to be aligned and follow the magnetic field, if there is any (h = 0).
The Gibbs measure for n spins is
γn(ω) = 1
Zn
j=1
e −βHn(ω) Pn(ω),
where Pn(ω) = 2 −n is the a priori measure (ω1, . . . , ωn are i.i.d. fair coin
flips), β > 0 is the inverse temperature, and Zn = e −βHn dPn is the normalization
constant called the partition function.
Remark 4.14. The more realistic Ising model in a finite box Λ ⊂ Zd has
Hamiltonian
H Ising
Λ (ω) = −J
ωxωy − h
ωx,
x,y:x−y 1 =1
where the summation is over x ∈ Λ and its nearest neighbors y. Curie-
Weiss is called the mean-field approximation of the Ising model because its
x