Gibbs measures and the Ising model
Let be a finite subset of and let . Let , a fixed configuration of spins outside . Let ; is the space of all configurations of spins on . We define a Hamiltonian (depending on the fixed external configuration ) by
gives the energy of a configuration , conditioned on the external configuration .
For a parameter (called the inverse temperature), we define the partition function by
Then we define the Gibbs distribution for the configuration space , depending on the external configuration , by
The purpose of the partition function is to normalize the above expression to be a probability measure on the configuration space .
For example, let be a square of side length centred at the origin, and take to be an external configuration of all negative spins. Define by
We show this configuration in Figure 1. We calculate that the energy of this configuration is . 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 square, there are nearest neighbor interactions. Put a line segment between every two lattice points with different spins; let be the set of these line segments. We show this in Figure 2.
Generally, if is an square then we have
Indeed, in our above example, and , so the above expression is , and we have already calculated that . What matters is that if we know the external configuration, then to describe the configuration inside a region it suffices to know the edges that separate opposite spins. And since the energy of any configuration has the term 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
is the set of contours corresponding to the configuration with the external configuration , and the summation is over all sets of nonintersecting contours.
We are not in fact interested in the Gibbs distribution on the configurations on a finite subset of , but instead limits of Gibbs distributions with . A Gibbs distribution on is in fact a probability measure on : for , a configuration on the plane, we define
Fix some . Let be a sequence of squares centred at the origin, let be a sequence of external configurations where all lattice points outside have positive spins, and let be a sequence of external configurations where all lattice points outside have negative spins. Let be the sequence of Gibbs distributions corresponding to the positive external spins, and let be the sequence of Gibbs distributions corresponding to the negative external spins. These extend to probability measures and on . Since is a compact metrizable space, the product is a compact metrizable space and thus the space of probability measures on it is compact. Hence the sequence has at least one limit point, say , and the sequence has at least one limit point, say . We shall show that , namely that there is not a unique limit Gibbs measure on the set of all configurations on .
Let and . Suppose that for all we had . Taking limits we have that and so (since the events and are disjoint and their union is the set of all configurations on ). But , so taking limits we also get . Therefore the measures and give different measures to the set , so they are distinct. Thus to show that the measures and are distinct it suffices to show that for all we have .
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 around the origin, . There are at most contours of length around the origin. Therefore
As , this is . In particular there is some such that if then for all we have . This shows that the limit Gibbs measures gives different measures to the set , hence they are distinct.
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