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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 \(\lambda = \lambda(x)\).


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 \(f\).


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