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Nearest Neighbour Interactions

In this section we consider Hamiltonians with \(V_k \equiv 0\) for \(k \geq 2\) and study the existence of the free energy, before analysing examples for particular choices of the potential \(V = V_1\). In this setting, we have

\begin{equation} H_N(u) = \sum_{\substack{i, j \in \Lambda \\ |i - j | = 1 }} V_1 \left( u_i - u_j \right)\end{equation}

Outline of the Method

The method we consider in this section works with symmetric potentials \(V\) which satisfy

\begin{equation}V\in C^{1}(\mathbb{R})\end{equation}

\begin{equation}M(\lambda):= \int_{\mathbb{R}} e^{\lambda\eta-V(\eta)}d\eta < \infty, \hspace{5mm} \forall\lambda\in\mathbb{R}\end{equation}

\begin{equation}\int_{\mathbb{R}} e^{\tau|V'(\eta)|-V(\eta)} d \eta < \infty, \hspace{5mm} \forall\tau\in\mathbb{R}\end{equation}

We assume that a limiting measure \(\mu^x\) exists for all \(x \in \mathbb R\), and will prove this in a special case (see report).

For \(\lambda \in \mathbb R\) we define the Cramér transform \(\hat{\nu}_\lambda \in \mathcal M^1( \mathbb R )\)

\begin{equation}\hat{\nu}_{\lambda}(d \eta):= \widehat{Z}_{\lambda}^{-1} \text{exp}{\left(\lambda\eta-V(\eta)\right)}d\eta\end{equation}

where \(\widehat Z_\lambda\) is the relevant partition function. We also define \(x(\lambda)\), the expected value of a bond under \(\hat \nu_\lambda\)

\begin{equation}x( \lambda ) := \mathbb{ E}_{\hat \nu_\lambda}[ \eta ] = \frac{d}{d \lambda} \log \widehat Z_{\lambda}.\end{equation}

It can be shown that \(x(\lambda)\) is strictly increasing and has a well defined inverse, denoted \(\lambda = \lambda(x)\).

Lemma

Let \(V: \mathbb R \rightarrow \mathbb R\) satisfy the above conditions and assume that a limiting gradient Gibbs measure, \(\mu^x\), exists for all tilts \(x \in \mathbb R\). Then the free energy \(f(x)\) exists, is differentiable, is strictly convex and satisfies
\begin{equation} f'(x) = \lambda(x).\end{equation}

(For proofs of results see report.)

The litrature concerning the results of this page predominantly use the term surface tension \(\sigma\) instead of free energy \(f\).


Examples

Next Nearest Neighbour Interactions