Gibbs measures and the Ising model

Let $\mathrm{\Lambda }$ be a finite subset of ${\mathbb{Z}}^2$ and let ${\mathrm{\Lambda }}^{\prime }={\mathbb{Z}}^2\setminus \mathrm{\Lambda }$. Let ${\sigma }^{\prime }\in {\{-1,+1\}}^{{\mathrm{\Lambda }}^{\prime }}$, a fixed configuration of spins outside $\mathrm{\Lambda }$. Let $\mathrm{\Omega }={\{-1,+1\}}^{\mathrm{\Lambda }}$; $\mathrm{\Omega }$ is the space of all configurations of spins on $\mathrm{\Lambda }$. We define a Hamiltonian $H_{\mathrm{\Lambda }}(\cdot |{\sigma }^{\prime }):\mathrm{\Omega }\to ℝ$ (depending on the fixed external configuration ${\sigma }^{\prime }$) by

$$H_{\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=-\sum _{\stackrel{x,y\in \mathrm{\Lambda }}{|x-y|=1}}\sigma (x)\sigma (y)-\sum _{\stackrel{x\in \mathrm{\Lambda },y\in {\mathrm{\Lambda }}^{\prime }}{|x-y|=1}}\sigma (x){\sigma }^{\prime }(y).$$

$H_{\mathrm{\Lambda }}(\cdot |{\sigma }^{\prime })$ gives the energy of a configuration $\sigma \in \mathrm{\Omega }$, conditioned on the external configuration ${\sigma }^{\prime }$.

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

$$Z(\beta ,\mathrm{\Lambda },{\sigma }^{\prime })=\sum _{\sigma \in \mathrm{\Omega }}\exp (-\beta H_{\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })).$$

Then we define the Gibbs distribution for the configuration space $\mathrm{\Omega }$, depending on the external configuration ${\sigma }^{\prime }$, by

$$P_{\beta ,\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=\frac{1}{Z(\beta ,\mathrm{\Lambda },{\sigma }^{\prime })}\exp (-\beta H(\sigma |{\sigma }^{\prime })).$$

The purpose of the partition function is to normalize the above expression to be a probability measure on the configuration space $\mathrm{\Omega }$.

For example, let $\mathrm{\Lambda }$ be a square of side length $3$ centred at the origin, and take ${\sigma }^{\prime }$ to be an external configuration of all negative spins. Define $\sigma \in \mathrm{\Omega }$ by

We show this configuration in Figure 1. We calculate that the energy of this configuration is $H_{\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=0$. 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 $n\times n$ square, there are $2n(n+1)$ nearest neighbor interactions. Put a line segment between every two lattice points with different spins; let $B(\sigma |{\sigma }^{\prime })$ 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 $\mathrm{\Lambda }$ is an $n\times n$ square then we have

$$H_{\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=-2n(n+1)+2|B(\sigma |{\sigma }^{\prime })|.$$

Indeed, in our above example, $n=3$ and $|B(\sigma |{\sigma }^{\prime })|=12$, so the above expression is $-24+2\cdot 12=0$, and we have already calculated that $H_{\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=0$. What matters is that if we know the external configuration, then to describe the configuration inside a region $\mathrm{\Lambda }$ 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

$$P_{\beta ,\mathrm{\Lambda }}(\sigma |{\sigma }^{\prime })=\frac{{\prod }_{\gamma \in \mathrm{\Gamma }(\sigma ,{\sigma }^{\prime })}\exp (-2|\gamma |)}{{\sum }_{\mathrm{\Gamma }}{\prod }_{\gamma \in \mathrm{\Gamma }}\exp (-2\beta |\gamma |)};$$

$\mathrm{\Gamma }(\sigma ,{\sigma }^{\prime })$ is the set of contours corresponding to the configuration $\sigma$ with the external configuration ${\sigma }^{\prime }$, and the summation is over all sets $\mathrm{\Gamma }$ of nonintersecting contours.

We are not in fact interested in the Gibbs distribution on the configurations on a finite subset $\mathrm{\Lambda }$ of ${\mathbb{Z}}^2$, but instead limits of Gibbs distributions with ${\mathrm{\Lambda }}_n\to {\mathbb{Z}}^2$. A Gibbs distribution $P_{\beta ,\mathrm{\Lambda }}(\cdot |{\sigma }^{\prime })$ on $\mathrm{\Omega }$ is in fact a probability measure on ${\{+1,-1\}}^{{\mathbb{Z}}^2}$: for $\sigma \in {\{+1,-1\}}^{{\mathbb{Z}}^2}$, a configuration on the plane, we define

Fix some $\beta$. Let ${\mathrm{\Lambda }}_n$ be a sequence of $n\times n$ squares centred at the origin, let ${\sigma }_{n,+}^{\prime }$ be a sequence of external configurations where all lattice points outside ${\mathrm{\Lambda }}_n$ have positive spins, and let ${\sigma }_{n,-}^{\prime }$ be a sequence of external configurations where all lattice points outside ${\mathrm{\Lambda }}_n$ have negative spins. Let $P_{n,+}$ be the sequence of Gibbs distributions corresponding to the positive external spins, and let $P_{n,-}$ be the sequence of Gibbs distributions corresponding to the negative external spins. These extend to probability measures ${\tilde{P}}_{n,+}$ and ${\tilde{P}}_{n,-}$ on ${\{+1,-1\}}^{{\mathbb{Z}}^2}$. Since $\{+1,-1\}$ is a compact metrizable space, the product ${\{+1,-1\}}^{{\mathbb{Z}}^2}$ is a compact metrizable space and thus the space of probability measures on it is compact. Hence the sequence ${\tilde{P}}_{n,+}$ has at least one limit point, say $P_{+}$, and the sequence ${\tilde{P}}_{n,-}$ has at least one limit point, say $P_{-}$. We shall show that $P_{+}\ne P_{-}$, namely that there is not a unique limit Gibbs measure on the set of all configurations on ${\mathbb{Z}}^2$.

Let $V_{+}=\{\sigma \in {\{+1,-1\}}^{{\mathbb{Z}}^2}:\sigma (0)=+1\}$ and $V_{-}=\{\sigma \in {\{+1,-1\}}^{{\mathbb{Z}}^2}:\sigma (0)=-1\}$. Suppose that for all $n$ we had . Taking limits we have that $P_{+}(V_{-})\le \frac{1}{3}$ and so $P_{+}(V_{+})\ge \frac{2}{3}$ (since the events $V_{+}$ and $V_{-}$ are disjoint and their union is the set of all configurations on ${\mathbb{Z}}^2$). But ${\tilde{P}}_{n,+}(V_{-})={\tilde{P}}_{n,-}(V_{+})$, so taking limits we also get $P_{-}(V_{+})\le \frac{1}{3}$. 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 .

We have

${\tilde{P}}_{n,+}(V_{-})$ $\le$ $\text{Prob}(\text{there exists a contour}\gamma \subset B(\sigma |{\sigma }^{\prime }),0\in \text{Int}(\gamma ))$

$\le$ ${\displaystyle \sum _{\stackrel{\gamma }{0\in \text{Int}(\gamma )}}}\text{Prob}(\gamma \subset B(\sigma |{\sigma }^{\prime }))$

$\le$ ${\displaystyle \sum _{\stackrel{\gamma }{0\in \text{Int}(\gamma )}}}\exp (-2\beta |\gamma |).$

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, $k\ge 4$. There are at most ${\left(\frac{k}{4}\right)}^2{4}^k$ contours of length $k$ around the origin. Therefore

$${\tilde{P}}_{n,+}(V_{-})\le \sum _{k=4}^{\mathrm{\infty }}\frac{k^2}{16}\cdot 4^k\exp (-2\beta k).$$

As $\beta \to \mathrm{\infty }$, this is $O(\exp (-8\beta ))$. In particular there is some ${\beta }_0$ such that if $\beta \ge {\beta }_0$ then for all $n$ we have . 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].