From 6c2b14dc9670a14589f9bff2420e375c3c11c01d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 16 Jun 2023 09:14:13 +0200
Subject: New figure and equation simplification and writing.

---
 when_annealed.tex | 73 ++++++++++++++++++++++++++++++++++++++-----------------
 1 file changed, 51 insertions(+), 22 deletions(-)

(limited to 'when_annealed.tex')

diff --git a/when_annealed.tex b/when_annealed.tex
index 90dc067..b92e9a5 100644
--- a/when_annealed.tex
+++ b/when_annealed.tex
@@ -25,7 +25,7 @@
 \begin{document}
 
 \title{
-  When is the average number of saddles typical?
+  When is the average number of saddles in a function typical?
 }
 
 \author{Jaron Kent-Dobias}
@@ -150,11 +150,15 @@ When the complexity is calculated using the Kac--Rice formula and a physicists'
 tool set, the problem is reduced to the evaluation of an integral by the saddle
 point method for large $N$ \cite{Kent-Dobias_2023_How}. An ansatz for
 the complexity needs to be made. Here we restrict ourselves to a
-{\oldstylenums1}\textsc{rsb} ansatz. We are guaranteed that
+{\oldstylenums1}\textsc{rsb} ansatz, which is parameterized by two quantities:
+$q_1$ and $x$. They have a geometric interpretation: given a stationary point
+fixed with certain properties, $1-x$ corresponds to the proportion of other
+stationary points with the same properties that are correlated with it, 
+and $q_1$ gives the overlap that this correlated population has with it. In the annealed or replica-symmetric case, $x=1$ and all but a vanishing fraction of stationary points are uncorrelated with each other.
+We are guaranteed that
 $\Sigma\leq\Sigma_{\oldstylenums1\textsc{rsb}}\leq\Sigma_\mathrm a$, and we
-will discuss later in what settings the {\oldstylenums1}\textsc{rsb}
-complexity is correct. The complexity is given by extremizing an effective
-action,
+will discuss later in what settings the {\oldstylenums1}\textsc{rsb} complexity
+is correct. The complexity is given by extremizing an effective action,
 \begin{equation}
   \Sigma_{\oldstylenums1\textsc{rsb}}(E,\mu)=\lim_{n\to0}\int dq_1\,dx\,\mathcal S(q_1,x\mid E,\mu)e^{nN\mathcal S(q_1,x\mid E,\mu)}
   =\mathop{\mathrm{extremum}}_{q_1,x}\mathcal S(q_1,x\mid E,\mu)
@@ -230,12 +234,16 @@ where we define for brevity (here and elsewhere) the constants
   u_f=f(f'+f'')-f'^2
   &&
   v_f=f'(f''+f''')-f''^2 \\
-  w_f=2f''^2+f'(f'''-2f'')
+  w_f=2f''(f''-f')+f'f'''
   &&
   y_f=f'(f'-f)+f''f
   &&
   z_f=f(f''-f')+f'^2
 \end{align}
+Note that for $f$ to define a sensible covariance, all of its series
+coefficients must be nonnegative. If $f$ has at least two nonzero coefficients
+at second order or higher, all of these constants are positive.
+
 Let $M$ be the matrix of double partial derivatives of $\mathcal S$ with
 respect to $q_1$ and $x$. We evaluate $M$ at the replica symmetric saddle point
 $x=1$ with the additional constraint that $q_1=1$ and along the extremal
@@ -243,14 +251,14 @@ complexity line \eqref{eq:extremal.line}. We determine when a zero eigenvalue
 appears, indicated the presence of a bifurcating {\oldstylenums1}\textsc{rsb}
 solution, by solving $0=\det M$. We find
 \begin{equation}
-  \det M=-\left(\frac{\partial^2\mathcal S}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\right)^2\propto(ay^2+bE^2+2cyE+d)^2
+  \det M=-\left(\frac{\partial^2\mathcal S}{\partial q_1\partial x}\bigg|_{\substack{x=1\\q_1=1}}\right)^2\propto(ay^2+bE^2+2cyE-d)^2
 \end{equation}
 where $y=-\frac12z_f\mu-f'f''E$ is proportional to the square-root term in \eqref{eq:extremal.line} and
 the constants $a$, $b$, $c$, and $d$ are defined below. Changing to $y$ is a
 convenient choice because the branch of \eqref{eq:extremal.line} is chosen
 by the sign of $y$ (the lower-energy branch we are interested in corresponds
 with $y>0$) and the relationship between $y$ and $E$ on the extremal line is
-$g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by
+$g=2hy^2+eE^2$, where the constants $e$, $g$, and $h$ are given by
 \begin{equation}
   \begin{aligned}
     a&=\frac{w_f\big(3y_f^2-4ff'f''(f'-f)\big)-6y_f^2(f''-f')f''}{(u_fz_ff'')^2f'}
@@ -259,7 +267,7 @@ $g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by
       \\
       c&=\frac{w_f}{f''z_f^2}
       \qquad
-      d=-\frac{w_f}{f'f''}
+      d=\frac{w_f}{f'f''}
       \qquad
       e=f''-f'
       \qquad
@@ -271,24 +279,20 @@ $g=2h^2y^2+eE^2$, where the constants $e$, $g$, and $h$ are given by
 The solutions for $\det M=0$ correspond to energies that satisfy
 \begin{equation}
   E_{\oldstylenums1\textsc{rsb}}^\pm
-  =-\operatorname{sign}[c(de+bg)]\frac{|c|g\mp\sqrt{c^2g^2-(2dh+ag)(de+bg)}}
+  =\operatorname{sign}(bg-de)\frac{-cg\pm\sqrt{c^2g^2+(2dh-ag)(bg-de)}}
   {
-    \sqrt{2c^2eg+(2bh-ae)(de+bg)\mp2|c|e\sqrt{c^2g^2-(2dh+ag)(de+bg)}}
+    \sqrt{2c^2eg+(2bh-ae)(bg-de)\mp2ce\sqrt{c^2g^2+(2dh-ag)(bg-de)}}
   }
 \end{equation}
 The expression inside the inner square root is proportional to
 \begin{equation}
-  \begin{aligned}
-    G_f
-    &=
-    -2(f''-f')u_fw_f
-    -2\log^2\frac{f''}{f'}f'^2f''v_f
-    \\
-    &\qquad
-    -f'\log\frac{f''}{f'}\Big[
-      4(f'-2f)(f''-f')f''^2-\big(-3(f'-f)f'^2+f'(f'-2f)f''+3ff''^2\big)f'''
-    \Big]
-  \end{aligned}
+  G_f
+  =
+  f'\log\frac{f''}{f'}\big[
+    3y_f(f''-f')f'''-2(f'-2f)f''w_f
+  \big]
+  -2(f''-f')u_fw_f
+  -2\log^2\frac{f''}{f'}f'^2f''v_f
 \end{equation}
 If $G_f>0$, then there are two points along the extremal complexity line where
 a solution bifurcates, and a new line of {\oldstylenums1}\textsc{rsb} solutions
@@ -343,6 +347,31 @@ between them. Therefore, $G_f>0$ is a necessary condition to see
   }
 \end{figure}
 
+\begin{figure}
+  \centering
+  \includegraphics{figs/order_plot_1.pdf}\\
+  \vspace{-1em}
+  \includegraphics{figs/order_plot_2.pdf}
+
+  \caption{
+    Examples of $3+14$ models where the solution
+    $E_{\oldstylenums1\textsc{rsb}}^-$ does and doesn't define the lower limit
+    of energies where \textsc{rsb} saddles are found. In both plots the red dot
+    shows $E_{\oldstylenums1\textsc{rsb}}^-$, while the solid red lines shows
+    the transition boundary with the \textsc{rs} complexity. The dashed black
+    line shows the \textsc{rs} zero complexity line, while the solid black line
+    shows the {\oldstylenums1}\textsc{rsb} zero complexity line. \textbf{Top:}
+    $\lambda=0.67$. Here the end of the transition line that begins at
+    $E_{\oldstylenums1\textsc{rsb}}^+$ does not match
+    $E_{\oldstylenums1\textsc{rsb}}^-$ but terminates at higher energies.
+    $E_{\oldstylenums1\textsc{rsb}}^-$ still corresponds with the lower bound.
+    \textbf{Bottom:} $\lambda=0.69$. Here the end of the transition line that
+    begins at $E_{\oldstylenums1\textsc{rsb}}^+$ terminates at lower energies
+    than $E_{\oldstylenums1\textsc{rsb}}^-$, and therefore its terminus defines
+    the lower bound.
+  }
+\end{figure}
+
 \begin{equation}
   \mu
   =-\frac{(f_1'+f_0'')u_f}{(2f_1-f_1')f_1'f_0''^{1/2}}
-- 
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