The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction

In the settings of The Real Numbers: Standing Notation and Background and The Flat Torus: Standing Notation, let $M\in\mathbb{N}$, and let $L$ and the lattice torus $\mathbb{L}_{M}$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. Let $\mathfrak{j}$ be a reflection-symmetric periodic pair interaction of side $L$ with interaction matrix $J$. Sums over finite index sets are those of Sum over a Finite Index Set, $\exp$ is the exponential function, and for a map $p:\mathcal{C}_{M}\to\mathbb{R}$ on the set $\mathcal{C}_{M}$ of clause 1 with $0\le p(\sigma)$ for every $\sigma\in\mathcal{C}_{M}$, $\lambda_{p}$ is the measure with point masses $p$ of Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure.

1. (Spin configurations)¶ A spin configuration is a map $\sigma:\mathbb{L}_{M}\to\{-1,1\}$; $\mathcal{C}_{M}=\mathrm{Map}(\mathbb{L}_{M},\{-1,1\})$ is the set of spin configurations, a nonempty finite set by The Maps from a Finite Set to a Nonempty Finite Set Form a Nonempty Finite Set §finite, $\mathbb{L}_{M}$ being nonempty and finite by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice and $\{-1,1\}$ having two elements. $2^{\mathcal{C}_{M}}$ is the power set of $\mathcal{C}_{M}$.

2. (The Hamiltonian)¶ The Hamiltonian of $\mathfrak{j}$ is the map $\mathcal{H}_{J}:\mathcal{C}_{M}\to\mathbb{R}$,

$$\mathcal{H}_{J}(\sigma)=-\frac{1}{2}\sum_{y\in\mathbb{L}_{M}}\sum_{z\in\mathbb{L}_{M}}J(y,z)\,\sigma(y)\,\sigma(z).$$

3. (The partition function)¶ The partition function of $\mathfrak{j}$ is the real number

$$Z_{J}=\sum_{\sigma\in\mathcal{C}_{M}}\exp\bigl(-\mathcal{H}_{J}(\sigma)\bigr),$$

which is positive, being a sum of positive terms over the nonempty finite set $\mathcal{C}_{M}$, each term positive by claim 2 of Basic Properties of the Exponential Function.

4. (The Ising measure)¶ The Ising measure of $\mathfrak{j}$ is $\mathbb{P}_{J}=\lambda_{p_{J}}$, the measure with point masses $p_{J}:\mathcal{C}_{M}\to\mathbb{R}$, $p_{J}(\sigma)=Z_{J}^{-1}\exp\bigl(-\mathcal{H}_{J}(\sigma)\bigr)$, which are positive. Since $\sum_{\sigma\in\mathcal{C}_{M}}p_{J}(\sigma)=Z_{J}^{-1}Z_{J}=1$, $\mathbb{P}_{J}$ is a probability measure on $(\mathcal{C}_{M},2^{\mathcal{C}_{M}})$ with $\mathbb{P}_{J}(\{\sigma\})=p_{J}(\sigma)>0$ for every $\sigma\in\mathcal{C}_{M}$, by Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy §measure.