# 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 ${H}_{\mathrm{\Lambda}}(\cdot |{\sigma}^{\prime}):\mathrm{\Omega}\to \mathbb{R}$ (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

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

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

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

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

$$\begin{array}{ccc}\sigma (-1,1)=+1\hfill & \sigma (0,1)=+1\hfill & \sigma (1,1)=-1\hfill \\ \sigma (-1,0)=-1\hfill & \sigma (0,0)=+1\hfill & \sigma (1,0)=-1\hfill \\ \sigma (-1,-1)=-1\hfill & \sigma (0,-1)=-1\hfill & \sigma (1,-1)=+1.\hfill \end{array}$$ |

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.

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

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

$\mathrm{\Gamma}(\sigma ,{\sigma}^{\prime})$ 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

$${\stackrel{~}{P}}_{\beta ,\mathrm{\Lambda}}(\sigma |{\sigma}^{\prime})=\{\begin{array}{cc}0\hfill & \sigma |{\mathrm{\Lambda}}^{\prime}\ne {\sigma}^{\prime}\hfill \\ {P}_{\beta ,\mathrm{\Lambda}}((\sigma |\mathrm{\Lambda})|{\sigma}^{\prime})\hfill & \sigma |{\mathrm{\Lambda}}^{\prime}={\sigma}^{\prime}.\hfill \end{array}$$ |

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 ${\stackrel{~}{P}}_{n,+}$ and ${\stackrel{~}{P}}_{n,-}$ 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 ${\stackrel{~}{P}}_{n,+}$ has at least one limit point, say ${P}_{+}$, and the sequence ${\stackrel{~}{P}}_{n,-}$ 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 $$. Taking limits we have that ${P}_{+}({V}_{-})\le \frac{1}{3}$ and so ${P}_{+}({V}_{+})\ge \frac{2}{3}$ (since the events ${V}_{+}$ and ${V}_{-}$ are disjoint and their union is the set of all configurations on ${\mathbb{Z}}^{2}$). But ${\stackrel{~}{P}}_{n,+}({V}_{-})={\stackrel{~}{P}}_{n,-}({V}_{+})$, so taking limits we also get ${P}_{-}({V}_{+})\le \frac{1}{3}$. Therefore the measures ${P}_{+}$ and ${P}_{-}$ give different measures to the set ${V}_{+}$, so they are distinct. Thus to show that the measures ${P}_{+}$ and ${P}_{-}$ are distinct it suffices to show that for all $n$ we have $$.

We have

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

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

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

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

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

As $\beta \to \mathrm{\infty}$, this is $O(\mathrm{exp}(-8\beta ))$. 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

## References

- [1] (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] (2009) More is the same; phase transitions and mean field theories. J. Stat. Phys. 137, pp. 777–797. Cited by: Further reading.
- [3] (2007) Introduction to (generalized) Gibbs measures. Note: arXiv:0712.1171 Cited by: Further reading.
- [4] (2000) Introduction to mathematical statistical physics. University Lecture Series, Vol. 19, American Mathematical Society, Providence, R.I.. Cited by: Further reading.
- [5] (1993) The statistical mechanics of lattice gases. Vol. I, Princeton University Press. Cited by: Further reading.
- [6] (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.