From 0277238fb24804b04a08934608ab262cb12339a9 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 2 Aug 2024 10:35:18 +0200
Subject: Some fixing.

---
 topology.tex | 36 ++++++++++++++++++++++++++++++------
 1 file changed, 30 insertions(+), 6 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 89692b7..097c5a4 100644
--- a/topology.tex
+++ b/topology.tex
@@ -298,9 +298,9 @@ solution manifold.
   \includegraphics{figs/bar.pdf}
 
   \caption{
-    Cartoon of the topology of the CCSP solution manifold implied by our
+    Cartoon of the topology of the solution manifold implied by our
     calculation. The arrow shows the vector $\mathbf x_0$ defining the height
-    function. The region of solutions is shaded orange, and the critical points
+    function. The region of solutions is marked in black, and the critical points
     of the height function restricted to this region are marked with a point.
     For $\alpha<1$, there are few simply connected regions with most of the
     minima and maxima contributing to the Euler characteristic concentrated at
@@ -371,15 +371,39 @@ consistent with the full RSB calculation of \cite{Urbani_2023_A}.
   &\quad-\frac\alpha2\log\det\big[I+\hat\beta\big(R\odot f'(C)\big)^{-1}f(C)\big]
 \end{align}
 
+The quenched average of the Euler characteristic in the replica symmetric ansatz becomes for $1<\alpha<\alpha_\text{\textsc{sat}}$
 \begin{align}
-  \mathcal S_0(m,r_d)
+  \frac1N\overline{\log\chi}
   =\frac12\bigg[
-    -m(m+r_d)+\log\left(-\frac m{r_d}\right) \notag \\
+    \log\left(-\frac 1{\tilde r_d}\right)
     -\alpha\log\left(
-      \frac{-m\big(f(1)-f(0)\big)+r_d\big(f'(1)-f(1)+f(0)\big)}{r_df'(1)}
+      1-\Delta f\frac{1+\tilde r_d}{f'(1)\tilde r_d}
     \right) \notag \\
-    +\frac{\alpha f(0)(m+r_d)}{-m\big(f(1)-f(0)\big)+r_d\big(f'(1)-f(1)+f(0)\big)}
+    -\alpha f(0)\left(\Delta f-\frac{f'(1)\tilde r_d}{1+\tilde r_d}\right)^{-1}
   \bigg]
 \end{align}
+where $\Delta f=f(1)-f(0)$ and $\tilde r_d$ is given by
+\begin{align}
+  \tilde r_d
+  =-\frac{f'(1)f(1)-\Delta f^2}{2(f'(1)-\Delta f)^2}
+  \bigg(
+    \alpha-2+\frac{2f'(1)f(0)}{f'(1)f(1)-\Delta f^2} \notag\\
+    +\sqrt{
+      \alpha^2
+      -4\alpha\frac{f'(1)f(0)\Delta f\big(f'(1)-\Delta f\big)}{\big(f'(1)f(1)-\Delta f^2\big)^2}
+    }
+  \bigg)
+\end{align}
+When $\alpha\to\alpha_\text{\textsc{sat}}=f'(1)/f(1)$ from below, $\tilde r_d\to -1$, which produces $N^{-1}\overline{\log\chi}\to0$.
+
+\begin{figure}
+  \includegraphics{figs/quenched.pdf}
+
+  \caption{
+    Comparison of the annealed and replica symmetric quenched calculations of
+    the logarithmic of the Euler characteristic. The covariance function is
+    $f(q)=\frac12+\frac12q^3$ and $\alpha_\text{\textsc{sat}}=\frac32$.
+  }
+\end{figure}
 
 \end{document}
-- 
cgit v1.2.3-70-g09d2

