From 36769c4221d3ee32993f14a937da7d039b8cb9d1 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 28 May 2023 12:21:45 +0200
Subject: Explaining the nearby expansion.

---
 2-point.tex | 30 ++++++++++++++++++++++++++----
 1 file changed, 26 insertions(+), 4 deletions(-)

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}
-- 
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