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diff --git a/2-point.tex b/2-point.tex index 1cc3687..d02fbfd 100644 --- a/2-point.tex +++ b/2-point.tex @@ -649,24 +649,46 @@ Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, \subsection{Most common neighbors with given overlap} +The most common neighbors of a reference point are given by further extremizing +the two-point complexity over the energy $E_1$ and stability $\mu_1$ of the +nearby points. This gives the conditions \begin{align} \hat\beta_1=0 && \mu_1=2r^{11}_\mathrm df''(1) \end{align} - +where the second is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the +nearby points are saddle points. Under the conditions where stationary points +can be found arbitrarily close to their neighbors, we can produce explicit +formulae for the complexity and the properties of the most common neighbors by +expanding in powers of $\Delta q=1-q$. For the complexity, the result is \begin{equation} \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q) +O\big((1-q)^2\big) \end{equation} +The popular of stationary points that are most common at each energy have the relation \begin{equation} - E_0^*=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0 + E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0 \end{equation} +between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this most common value, the energy and stability of the most common neighbors at small $\Delta q$ are \begin{equation} - \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}(E_0-E_0^*)(1-q)+O\big((1-q)^2\big) + E_1=E_0+\frac12\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \end{equation} \begin{equation} - E_1=E_0+\frac12\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}(E_0-E_0^*)(1-q)^2+O\big((1-q)^3\big) + \mu_1=\mu_0-\frac{f'(1)(f'''(1)+f''(1))-f''(1)^2}{f(1)(f'(1)+f''(1))-f'(1)^2}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big) \end{equation} +Therefore, whether the energy and stability of nearby points increases or +decreases from that of the reference point depends only on whether the energy +of the reference point is above or below that of the most common population at +the same stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm +m)=E_\mathrm{th}$, the threshold energy is also the pivot around which the +points asymptotically nearby marginal minima change their properties. + +To examine better the population of marginal points, it is necessary to look at +the next term in the series of the complexity with $\delta q$, since the linear +coefficient becomes zero at the marginal line. This tells us something +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. \section{Isolated eigenvalue} |