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