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diff --git a/2-point.tex b/2-point.tex index efc7a71..f8d700a 100644 --- a/2-point.tex +++ b/2-point.tex @@ -743,6 +743,18 @@ intuitive: stable minima have an effective repulsion between points, and one always finds a sufficiently small $\Delta q$ that no stationary points are point any nearer. For the marginal minima, it is not clear that the same should be true. +When $\mu=\mu_\mathrm m$, the linear term above vanishes. Under these conditions, the quadratic term in the expansion is +\begin{equation} + \Sigma_{12} + =\frac12\frac{f'''(1)\big(f'(1)(f''(1)+f'''(1))-f''(1)^2\big)}{f''(1)^{3/2}\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)} + \left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)\big(E_0-E_\textrm{th}\big)(1-q)^2+O\big((1-q)^3\big) +\end{equation} +Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore, +among marginal minima, when $E_0$ is greater than the threshold one finds +neighbors at arbitrarily close distance. When $E_0$ is less than the threshold, +the complexity of nearby points is negative, and there is a desert where none +are found. + \section{Isolated eigenvalue} \label{sec:eigenvalue} |