From 6da88b5a8251a59472bc3d16c9cc0a6ac6efb8c8 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 7 Jun 2023 23:23:51 +0200
Subject: Some writing.

---
 2-point.tex | 62 ++++++++++++++++++++++++++++++++++++++-----------------------
 1 file changed, 39 insertions(+), 23 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index 6cf2f60..9d951e9 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -244,6 +244,8 @@ and a $3+8$ model tuned to maximize the ``interesting'' region of the dynamics r
 \end{equation}
 
 \begin{figure}
+  \centering
+  \includegraphics{figs/single_complexity.pdf}
   \caption{
     Plots of the complexity (logarithm of the number of stationary points) for
     the mixed spherical models studied in this paper. Energies and stabilities
@@ -626,15 +628,24 @@ they take when the ordinary, 1-point complexity is calculated. For a replica
 symmetric complexity of the reference point, this results in
 \begin{align}
   \hat\beta_0
-  &=-\frac{(E_0+\mu_0)f'(1)+E_0f''(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}\\
+  &=-\frac{\mu_0f'(1)+E_0\big(f'(1)+f''(1)\big)}{u_f}\\
   r_\mathrm d^{00}
-  &=\frac{\mu_0f(1)+E_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2} \\
+  &=\frac{\mu_0f(1)+E_0f'(1)}{u_f} \\
   d_\mathrm d^{00}
   &=\frac1{f'(1)}
   -\left(
-    \frac{\mu_0f(1)+E_0f'(1)}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}
+    \frac{\mu_0f(1)+E_0f'(1)}{u_f}
   \right)^2
 \end{align}
+where we define for brevity (here and elsewhere) the constants
+\begin{align}
+  u_f=f(1)\big(f'(1)+f''(1)\big)-f'(1)^2
+  &&
+  v_f=f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2
+\end{align}
+Note that because the coefficients of $f$ must be nonnegative for $f$ to
+be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. In
+general, $f^{(n)}(1)\geq f^{(m)}(1)$ if $n>m$.
 
 
 In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
@@ -764,7 +775,7 @@ $r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is
   \end{aligned}
 \end{equation}
 
-\subsection{Most common neighbors with given overlap}
+\subsection{Expansion in the near neighborhood}
 
 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
@@ -788,9 +799,9 @@ The population of stationary points that are most common at each energy have the
 \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{align} \label{eq:expansion.E.1}
-  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) \\
+  E_1&=E_0+\frac12\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\
     \label{eq:expansion.mu.1}
-  \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)
+  \mu_1&=\mu_0-\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big)
 \end{align}
 Therefore, whether the energy and stability of nearby points increases or
 decreases from that of the reference point depends only on whether the energy
@@ -809,7 +820,7 @@ point any nearer. For the marginal minima, it is not clear that the same should
 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)}
+  =\frac12\frac{f'''(1)v_f}{f''(1)^{3/2}u_f}
   \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,
@@ -825,24 +836,29 @@ $\mu_1=\mu_0+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
 where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1}
 and
 \begin{equation}
-  \delta\mu_2=\frac{f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2}{f'(1)f''(1)^{3/4}}\sqrt{
-    \frac{E_0-E_\mathrm{th}}2\frac{f'(1)\big(f'''(1)-2f''(1)\big)+2f''(1)^2}{f(1)\big(f'(1)+f''(1)\big)-f'(1)^2}
+  \delta\mu_2=\frac{v_f}{f'(1)f''(1)^{3/4}}\sqrt{
+    \frac{E_0-E_\mathrm{th}}2\frac{f'(1)\big(f'''(1)-2f''(1)\big)+2f''(1)^2}{u_f}
   }
 \end{equation}
-Similarly, one finds that the energy lies in the range $E_1=E_0+\delta E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given by the coefficient in \eqref{eq:expansion.E.1} and
+Similarly, one finds that the energy lies in the range $E_1=E_0+\delta
+E_1(1-q)^2\pm\delta E_2(1-q)^{5/2}+O\big((1-q)^3\big)$ for $\delta E_1$ given
+by the coefficient in \eqref{eq:expansion.E.1} and
 \begin{equation}
-  \delta E_2
-  =\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\sqrt{
-    \frac{
-      \big[f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2\big]\big[
-        f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)
-      \big]
-      \big[
-        f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))
-      \big]
-    }
-    {3\big(f(1)(f'(1)+f''(1))-f'(1)^2\big)}
-  }
+  \begin{aligned}
+    \delta E_2
+    &=\frac{\sqrt{E_0-E_\mathrm{th}}}{4f'(1)f''(1)^{3/4}}\bigg(
+      \frac{
+        v_f
+        }{3u_f}
+        \big[
+          f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))
+        \big]
+        \\
+    &\hspace{17pc}\times
+        \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)
+        \big]
+    \bigg)^\frac12
+  \end{aligned}
 \end{equation}
 and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1-q)^2)$ and is given by the real root to the quintic equation
 \begin{equation}
@@ -998,7 +1014,7 @@ which the Hessian is evaluated.
   }
 \end{figure}
 
-Using the same methodology as above, the disorder-dependant terms are captured in the linear operator
+Using the same methodology as above, the disorder-dependent terms are captured in the linear operator
 \begin{equation}
   \mathcal O(\mathbf t)=
   \sum_a^m\delta(\mathbf t-\pmb\sigma_a)(i\hat{\pmb\sigma}_a\cdot\partial_\mathbf t-\hat\beta_0)
-- 
cgit v1.2.3-54-g00ecf

