# Gibbs measures and the Ising model

Jordan Bell
July 1, 2014

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

 $H_{\Lambda}(\sigma|\sigma^{\prime})=-\sum_{\stackrel{{\scriptstyle x,y\in% \Lambda}}{{|x-y|=1}}}\sigma(x)\sigma(y)-\sum_{\stackrel{{\scriptstyle x\in% \Lambda,y\in\Lambda^{\prime}}}{{|x-y|=1}}}\sigma(x)\sigma^{\prime}(y).$

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

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

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

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

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

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

For example, let $\Lambda$ 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 $\sigma\in\Omega$ by

 $\begin{array}[]{lll}\sigma(-1,1)=+1&\sigma(0,1)=+1&\sigma(1,1)=-1\\ \sigma(-1,0)=-1&\sigma(0,0)=+1&\sigma(1,0)=-1\\ \sigma(-1,-1)=-1&\sigma(0,-1)=-1&\sigma(1,-1)=+1.\end{array}$

We show this configuration in Figure 1. We calculate that the energy of this configuration is $H_{\Lambda}(\sigma|\sigma^{\prime})=0$. 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.

Generally, if $\Lambda$ is an $n\times n$ square then we have

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

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 $H_{\Lambda}(\sigma|\sigma^{\prime})=0$. What matters is that if we know the external configuration, then to describe the configuration inside a region $\Lambda$ 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

 $P_{\beta,\Lambda}(\sigma|\sigma^{\prime})=\frac{\prod_{\gamma\in\Gamma(\sigma,% \sigma^{\prime})}\exp(-2|\gamma|)}{\sum_{\Gamma}\prod_{\gamma\in\Gamma}\exp(-2% \beta|\gamma|)};$

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

We are not in fact interested in the Gibbs distribution on the configurations on a finite subset $\Lambda$ of $\mathbb{Z}^{2}$, but instead limits of Gibbs distributions with $\Lambda_{n}\to\mathbb{Z}^{2}$. A Gibbs distribution $P_{\beta,\Lambda}(\cdot|\sigma^{\prime})$ on $\Omega$ 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

 $\widetilde{P}_{\beta,\Lambda}(\sigma|\sigma^{\prime})=\begin{cases}0&\sigma|% \Lambda^{\prime}\neq\sigma^{\prime}\\ P_{\beta,\Lambda}((\sigma|\Lambda)|\sigma^{\prime})&\sigma|\Lambda^{\prime}=% \sigma^{\prime}.\end{cases}$

Fix some $\beta$. Let $\Lambda_{n}$ 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 $\Lambda_{n}$ have positive spins, and let $\sigma_{n,-}^{\prime}$ be a sequence of external configurations where all lattice points outside $\Lambda_{n}$ 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 $\widetilde{P}_{n,+}$ and $\widetilde{P}_{n,-}$ 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 $\widetilde{P}_{n,+}$ has at least one limit point, say $P_{+}$, and the sequence $\widetilde{P}_{n,-}$ has at least one limit point, say $P_{-}$. We shall show that $P_{+}\neq P_{-}$, 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 $\widetilde{P}_{n,+}(V_{-})<\frac{1}{3}$. Taking limits we have that $P_{+}(V_{-})\leq\frac{1}{3}$ and so $P_{+}(V_{+})\geq\frac{2}{3}$ (since the events $V_{+}$ and $V_{-}$ are disjoint and their union is the set of all configurations on $\mathbb{Z}^{2}$). But $\widetilde{P}_{n,+}(V_{-})=\widetilde{P}_{n,-}(V_{+})$, so taking limits we also get $P_{-}(V_{+})\leq\frac{1}{3}$. 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 $\widetilde{P}_{n,+}(V_{-})<\frac{1}{3}$.

We have

 $\displaystyle\widetilde{P}_{n,+}(V_{-})$ $\displaystyle\leq$ $\displaystyle\text{Prob}\left(\textrm{there exists a contour}\,\gamma\subset B% (\sigma|\sigma^{\prime}),0\in\text{Int}(\gamma)\right)$ $\displaystyle\leq$ $\displaystyle\sum_{\stackrel{{\scriptstyle\gamma}}{{0\in\text{Int}(\gamma)}}}% \text{Prob}(\gamma\subset B(\sigma|\sigma^{\prime}))$ $\displaystyle\leq$ $\displaystyle\sum_{\stackrel{{\scriptstyle\gamma}}{{0\in\text{Int}(\gamma)}}}% \exp(-2\beta|\gamma|).$

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

 $\widetilde{P}_{n,+}(V_{-})\leq\sum_{k=4}^{\infty}\frac{k^{2}}{16}\cdot 4^{k}% \exp(-2\beta k).$

As $\beta\to\infty$, this is $O(\exp(-8\beta))$. In particular there is some $\beta_{0}$ such that if $\beta\geq\beta_{0}$ then for all $n$ we have $\widetilde{P}_{n,+}(V_{-})<\frac{1}{3}$. This shows that the limit Gibbs measures gives different measures to the set $V_{+}$, hence they are distinct.