Gibbs measures and the Ising model

Jordan Bell
July 1, 2014

Let Λ be a finite subset of 2 and let Λ=2Λ. Let σ{-1,+1}Λ, a fixed configuration of spins outside Λ. Let Ω={-1,+1}Λ; Ω is the space of all configurations of spins on Λ. We define a Hamiltonian HΛ(|σ):Ω (depending on the fixed external configuration σ) by


HΛ(|σ) gives the energy of a configuration σΩ, conditioned on the external configuration σ.

For a parameter β>0 (called the inverse temperature), we define the partition function by


Then we define the Gibbs distribution for the configuration space Ω, depending on the external configuration σ, by


The purpose of the partition function is to normalize the above expression to be a probability measure on the configuration space Ω.

For example, let Λ be a square of side length 3 centred at the origin, and take σ to be an external configuration of all negative spins. Define σΩ by


We show this configuration in Figure 1. We calculate that the energy of this configuration is HΛ(σ|σ)=0. We can calculate the energy of this configuration in a different way, using line segments separating lattice points with different spins, as follows. For an n×n square, there are 2n(n+1) nearest neighbor interactions. Put a line segment between every two lattice points with different spins; let B(σ|σ) be the set of these line segments. We show this in Figure 2.

Figure 1: An example of a configuration (and negative external spins)
Figure 2: Calculating energy using contours

Generally, if Λ is an n×n square then we have


Indeed, in our above example, n=3 and |B(σ|σ)|=12, so the above expression is -24+212=0, and we have already calculated that HΛ(σ|σ)=0. What matters is that if we know the external configuration, then to describe the configuration inside a region Λ it suffices to know the edges that separate opposite spins. And since the energy of any configuration has the term -2n(n+1) and this appears in the numerator and denominator of the expression for the Gibbs distribution, we can omit it to calculate the Gibbs distribution. By a contour we mean a closed path of edges that does not intersect itself. We can express the Gibbs distribution in terms of contours as


Γ(σ,σ) is the set of contours corresponding to the configuration σ with the external configuration σ, and the summation is over all sets Γ of nonintersecting contours.

We are not in fact interested in the Gibbs distribution on the configurations on a finite subset Λ of 2, but instead limits of Gibbs distributions with Λn2. A Gibbs distribution Pβ,Λ(|σ) on Ω is in fact a probability measure on {+1,-1}2: for σ{+1,-1}2, a configuration on the plane, we define


Fix some β. Let Λn be a sequence of n×n squares centred at the origin, let σn,+ be a sequence of external configurations where all lattice points outside Λn have positive spins, and let σn,- be a sequence of external configurations where all lattice points outside Λn have negative spins. Let Pn,+ be the sequence of Gibbs distributions corresponding to the positive external spins, and let Pn,- be the sequence of Gibbs distributions corresponding to the negative external spins. These extend to probability measures P~n,+ and P~n,- on {+1,-1}2. Since {+1,-1} is a compact metrizable space, the product {+1,-1}2 is a compact metrizable space and thus the space of probability measures on it is compact. Hence the sequence P~n,+ has at least one limit point, say P+, and the sequence P~n,- has at least one limit point, say P-. We shall show that P+P-, namely that there is not a unique limit Gibbs measure on the set of all configurations on 2.

Let V+={σ{+1,-1}2:σ(0)=+1} and V-={σ{+1,-1}2:σ(0)=-1}. Suppose that for all n we had P~n,+(V-)<13. Taking limits we have that P+(V-)13 and so P+(V+)23 (since the events V+ and V- are disjoint and their union is the set of all configurations on 2). But P~n,+(V-)=P~n,-(V+), so taking limits we also get P-(V+)13. Therefore the measures P+ and P- give different measures to the set V+, so they are distinct. Thus to show that the measures P+ and P- are distinct it suffices to show that for all n we have P~n,+(V-)<13.

We have

P~n,+(V-) Prob(there exists a contourγB(σ|σ),0Int(γ))

The above sum is over all contours such that the origin lies in their interior. We can write the set of all contours around the origin as a union of the set of all contours of length k around the origin, k4. There are at most (k4)24k contours of length k around the origin. Therefore


As β, this is O(exp(-8β)). In particular there is some β0 such that if ββ0 then for all n we have P~n,+(V-)<13. This shows that the limit Gibbs measures gives different measures to the set V+, hence they are distinct.

Further reading

Minlos [4], Sinai [6], Cipra [1], Simon [5], Le Ny [3], Kadanoff [2].


  • [1] B. A. Cipra (1987) An introduction to the Ising model. Amer. Math. Monthly 94 (10), pp. 937–959. External Links: ISSN 0002-9890, Document, Link, MathReview (Peter J. Forrester) Cited by: Further reading.
  • [2] L. P. Kadanoff (2009) More is the same; phase transitions and mean field theories. J. Stat. Phys. 137, pp. 777–797. Cited by: Further reading.
  • [3] A. Le Ny (2007) Introduction to (generalized) Gibbs measures. Note: arXiv:0712.1171 Cited by: Further reading.
  • [4] R. A. Minlos (2000) Introduction to mathematical statistical physics. University Lecture Series, Vol. 19, American Mathematical Society, Providence, R.I.. Cited by: Further reading.
  • [5] B. Simon (1993) The statistical mechanics of lattice gases. Vol. I, Princeton University Press. Cited by: Further reading.
  • [6] Ya. G. Sinai (1982) Theory of phase transitions: rigorous results. Pergamon Press, Oxford. Note: Translated from the Russian External Links: ISBN 0-08-026469-7, MathReview (Gunduz Caginalp) Cited by: Further reading.