# Examples

#### Gaussian Potential

A Gaussian potential has the form

\begin{equation}V(\eta) = \frac{c}{2}\eta^{2},\quad c > 0.\end{equation}

For this choice of potential the partition function \(\widehat Z_\lambda\) can be computed explicitly as a Gaussian integral

\[\widehat Z_\lambda = \int_{\mathbb R} e^{\lambda \eta - \frac{c}{2} \eta^2} d\eta = \sqrt{ \frac {2\pi}{c} } e^{\lambda^2/ 2c }.\]

Then by the definition of \(x(\lambda)\), taking logs and differentiating gives \(x(\lambda) = \lambda / c\); it follows that \(\lambda(x) = c x\) and hence

\begin{equation}f(x) = \frac{c}{2}x^2.\end{equation}

### Superposition of Gaussian Potentials

A non-convex example for which our method may be used is:

\begin{equation}\exp \big( - V(\eta) \big) = p \exp \left( - \frac{\kappa_1}{2} \eta^2 \right) + (1-p) \exp \left( - \frac{\kappa_2}{2}\eta^2 \right).\end{equation}

Here \(\kappa_1, \kappa_2>0\) are stiffness parameters and it is generally assumed that \(\kappa_1 \gg \kappa_2\). The measure can be reformulated so that with probability \(p\) (respectively \(1-p\)) the distribution is conditioned to behave according to the density \(\exp( - \frac{\kappa_1}{2} \eta^2 )\) ( respectively \(\exp( - \frac{\kappa_2}{2} \eta^2 )\)).

Using the same Gaussian integral evaluation we used for the potential \(V(\eta) = \frac{c}{2}\eta^{2}\), we can compute the partition function \(\widehat Z_\lambda\) for the superposed Gaussian potential

\begin{equation}\hat {Z_{\lambda}} = p \sqrt{ \frac{2 \pi}{\kappa_1}} e^{\lambda^2 / 2 \kappa_1} + (1-p) \sqrt{ \frac{2 \pi}{\kappa_2}} e^{\lambda^2 / 2 \kappa_2}.\end{equation}

Again drawing comparison with the simple Gaussian potential, we identify \(x(\lambda)\)

\begin{equation}x(\lambda) = \left(\frac{\kappa_2a+\kappa_1b}{\kappa_1\kappa_2(a+b)}\right) \lambda\end{equation}

where \(a:= pe^{\frac{\lambda^{2}}{2\kappa_1}}\sqrt{\frac{2\pi}{\kappa_1}}\) and \(b:=(1-p) e^{\frac{\lambda^{2}}{2\kappa_2}}\sqrt{\frac{2\pi}{\kappa_2}}\), and we define

\[C(\lambda):=\left(\frac{\kappa_2a+\kappa_1b}{\kappa_1\kappa_2(a+b)}\right),\]

so that \(x(\lambda) = C(\lambda) \lambda\). If we choose the stiffness parameters so that \(\kappa_1, \kappa_2 > 1\) then \((\kappa_1 \kappa_2 )^{-1} < C(\lambda) < 1\); from this we obtain bounds on the free energy

\begin{equation} \frac{x^{2}}{2} < f(x) < \kappa_1\kappa_2\frac{x^{2}}{2}.\end{equation}

Similarly when \(\kappa_1, \kappa_2 < 1\) we have

\[\kappa_1\kappa_2\frac{x^{2}}{2} < f(x) < \frac{x^{2}}{2}.\]

Whilst we have not been able to obtain a formula for \(f\), our method has enabled us to find bounds which show that of \(f\) is \(O(x)\).

### Double Well Potential

As a final example we consider the double well potential

\begin{equation}V(\eta) := (\eta^{2}-1)^{2}.\end{equation}

Applying our main Lemma we note that the free energy exists and is strictly convex. Whilst we have **not found an explicit formula** for the free energy, the fact that it is strictly convex is interesting in itself, since one would not expect this of a potential that has two global energy minima.

For such a potential one would expect to find two distinct equilibrium configurations, one of which favours \(+1\)-valued bonds, whilst the other has a majority of \(-1\)-valued bonds, in turn corresponding to the existence of two distinct Gibbs measures, and the occurrence of a phase transition. However, the strict convexity of the surface tension corresponds to a strict energy minimum of the system, which would appear to contradict this reasoning, and indicates (though does not prove) a lack of phase transition.

This somewhat counter intuitive situation is commonplace in \(1\)-dimension, and can be partially explained by comparison to other models. Noting that the potential \(V\) has energy minima at \(\pm 1\) and has ground states of pure \(+1\) and pure \(-1\) configurations, it is somewhat natural to relate it to the \(1\)-dimensional **Ising model**, which is known not to exhibit a phase transition even though the model does in higher dimensions.

A second comparison is to the **random walk,** with distribution described by the potential \(V\). In \(1\)-dimension, the random walk is known to fluctuate wildly, and hence is unlikely to remain in either of the states favoured by the distribution.

As a final justification, consider the case of** zero tilt** (purely for ease of understanding), \(x = 0\). The loop condition asserts that the mean value taken by a bond is \(0\), and since the double well concentrates mass around bonds taking \(\pm 1\) values, this is saying that we expect there to be equally as many \(+1\) bond as their are \(-1\). Taking the thermodynamic limit, one expects to preserve this balance between positive and negative bonds, which is to say that there is a unique limiting Gibbs measure in which we have phase coexistence, rather than the expected distinct Gibbs measures corresponding to the two possible phases.