From 910a0d4f968809bb6ab9ac7731ce84df9855371d Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 8 Sep 2024 16:13:09 +0200
Subject: Started typing details of correct solution for the complexity.

---
 topology.tex | 119 ++++++++++++++++++++++++++++++++---------------------------
 1 file changed, 64 insertions(+), 55 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 212a20d..ee658a0 100644
--- a/topology.tex
+++ b/topology.tex
@@ -1391,78 +1391,87 @@ The effective action then has the form
 \end{equation}
 It is not helpful to write out this entire expression, which is quite large.
 However, we only find saddle points of this action with $\bar
-J=J=G_{12}=G_{21}=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$. Setting these variables to zero, we find
+J=J=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$ and $G_{21}=G_{12}$. Setting these variables to zero, we find
 \begin{align}
   \notag
   \mathcal S
   =\hat m
-  +\frac12i\epsilon A_-
   -\alpha\frac12\log\frac{
-    f'(1)(Df(1)+R^2f'(1))-\frac12((R-\frac12A_+)^2+\frac12 A_-^2)f(1)f''(1)
+    f'(1)(Df(1)+R^2f'(1))-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f(1)f''(1)
   }{
-    \frac14(R+\frac12A_+)^2f'(1)^2
+    [\frac12(R+A_+)^2-2G_{12}^2]f'(1)^2
   }
   \\
   \notag
-  -\alpha\frac12\log\frac{
-    \epsilon^2+i\epsilon f'(1)A_-+\frac14(A_+^2-A_-^2)f'(1)^2
+  +\frac12(1-\alpha)\log\frac{
+    4(A_+^2-A_-^2)
   }{
-    \frac14(R+\frac12A_+)^2f'(1)^2
+    \frac12(R+A_+)^2-2G_{12}^2
+  }
+  +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{
+    \frac12(R+A_+)^2-2G_{12}^2
   }
-  +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{\frac14(R+\frac12A_+)^2}
   \\
-  +\frac12\log\frac{A_+^2-A_-^2}{(R+\frac12A_+)^2}
   -\frac12\alpha V_0^2\left(
-    f(1)+\frac{R^2f'(1)^2}{Df'(1)-\frac12((R-\frac12A_+)^2+\frac12A_-^2)f''(1)}
+    f(1)+\frac{R^2f'(1)^2}{Df'(1)-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f''(1)}
   \right)^{-1}
 \end{align}
-One solution to these equations at $\epsilon=0$ is $A_-=0$ and $A_+=2R^*$ with
-$D=\hat m=0$ and $R=R^*$, exactly as for the Euler characteristic. The resulting effective
-action as a function of $m$ is also exactly the same. We find two other
-solutions that differ from this one, but with formulae too complex to share in
-the text. One alternate solution has $A_-=0$ and one has $A_-\neq0$.
-
-\begin{figure}
-  \includegraphics{figs/alt_sols_1.pdf}
-  \hfill
-  \includegraphics{figs/alt_sols_2.pdf}
+One solution to these equations is $A_-=G_{12}=0$ and $A_+=R^*$ with
+$D$, $\hat m$, and $R$ exactly as for the Euler characteristic. The resulting effective
+action as a function of $m$ is also exactly the same. There is another solution, this time with $A_+=R$, $A_-=G_{12}$, and
+\begin{align}
+  &R=\frac{\alpha mf'(1)}{b^2}\big[
+    V_0^2f'(1)^2(1-m^2)-f(1)b
+  \big]
+  \\
+  &G_{12}^2
+  =\frac{f'(1)mR}{f''(1)(1-m^2)b^2}
+  \big[\alpha V_0^2f'(1)^2f''(1)(1-m^2)^2-b^2-\alpha bf'(1)(f(1)-(1-m^2)f'(1))
+  \big]
+\end{align}
+where we have defined the constant
+\begin{equation}
+  b=(1-m^2)[f''(1)f(1)+f'(1)^2]-f(1)f'(1)
+\end{equation}
+The resulting form for the action is
+\begin{align}
+  \mathcal S_\mathcal N(m)
+  =\frac12(1-\alpha)\log\left(
+    \frac{f''(1)(1-m^2)}{f'(1)(1-\alpha)}
+  \right)
+  -\frac12\alpha\log\left(
+    \frac{\alpha f'(1)V_0^2}b
+  \right)
+  -\frac12\alpha\frac{V_0^2\big[f''(1)(1-m^2)-f'(1)\big]+b}b
+\end{align}
+This solution is plotted alongside the solution that coincides with that of the
+Euler characteristic in Fig. It is clear from this plot that the new solution
+cannot be valid in the entire range of $m$, since it diverges as $m$ goes to 1
+where we know there are vanishingly few stationary points. However, there is a
+single point $m_c$ where the two solutions coincide, and they have the
+possibility of trading stability. This is given by
+\begin{equation}
+  m_c^2
+  =1-\frac{2(1-\alpha)f(1)f'(1)}{
+    (2-\alpha)f(1)f''(1)+2(1-\alpha)f'(1)^2
+    -\sqrt{\alpha f''(1)}\sqrt{4V_0^2(1-\alpha)f'(1)^2+\alpha f(1)^2f''(1)}
+  }
+\end{equation}
 
-  \caption{
-    \textbf{Choosing the correct saddle-point for the complexity.} In the
-    calculation of the complexity of stationary points in the height function,
-    there are three plausible solutions in some regimes. \textbf{Left:} Plot of
-    the three solutions for $M=1$, $f(q)=\frac12q^3$, and
-    $E=E_\text{on}=3^{-1/2}$. The inset shows detail of the region where these
-    solutions are nonnegative. The solid and dashed lines show the values of
-    $m$ where the $A_-=0$ and $A_-\neq0$ alternative solutions are maximized,
-    respectively. The value $m^*$ maximizing the action corresponding also to
-    $\mathcal S_\chi$ is also marked. \textbf{Right:} Numeric measurement of the
-    average value of $m$ for stationary points found using Newton's method as a
-    function of model size $N$. The thick solid line is a power-law fit to
-    $m-m^*$, while the thin solid and dashed lines show the same values of $m$
-    as in the lefthand plot.
-  } \label{fig:alt.sols}
-\end{figure}
+\begin{equation}
+  \mathcal S_\mathcal N(m)
+  =\begin{cases}
+    \mathcal S_\chi(m) & m^2\geq m_c^2 \\
+    \mathcal S_{\ast}(m) & m^2<m_c^2
+  \end{cases}
+\end{equation}
+This formula remains valid also in the regime when $m_c^2<0$, when the
+complexity is given by the function $\mathcal S_\chi$ in the entire range
+$m\in(-1,1)$.
 
-For both alternative solutions, there is a regime of small $V_0$ or $E$ for
-which they are unphysical. In this case the action is everywhere negative
-despite our knowledge from the calculation of the Euler characteristic that the
-manifold exists. However, there is a second regime where they cross the action
-$\mathcal S_\chi$ and become positive, both reaching values of $m$ that are
-less than $m_\text{min}$. An example of this regime for the case with
-$f(q)=\frac12q^3$ and $M=1$ (the spherical spin glass) is shown in the left
-panel of Fig.~\ref{fig:alt.sols}. In order to investigate which solution is
-valid in this regime, we turn to comparison with numeric experiments. We create
-samples of this model at the specified parameters, choose random axis $\mathbf
-x_0$, and use Newton's method with mild damping to find stationary points of
-the Lagrangian starting from random initial conditions. When we find a
-stationary point, we record the value of the overlap $m=\frac1N\mathbf
-x\cdot\mathbf x_0$ between it and the height axis. The right panel of
-Fig.~\ref{fig:alt.sols} shows that the average value of $m$ attained by this
-method as a function of $N$ asymptotically approaches $m^*$, the value implied
-by the Euler characteristic action $\mathcal S_\chi$, and not either of the
-values implied by the alternative solutions. This indicates that $\mathcal
-S_\mathcal N(m)=\mathcal S_\chi(m)$ in all regimes.
+\begin{equation}
+  V_\text{on}^2=\frac{(1-\alpha)f'(1)^2-\alpha f(1)f''(1)}{\alpha f''(1)}
+\end{equation}
 
 \section{The quenched shattering energy}
 \label{sec:1frsb}
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