From cd9302e207922e62bd223f5dfa6ac1f1bca1db62 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 18 Sep 2024 13:57:15 +0200
Subject: Some small fixes.

---
 topology.tex | 13 +++++++++----
 1 file changed, 9 insertions(+), 4 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 8ff0de1..3b6afc1 100644
--- a/topology.tex
+++ b/topology.tex
@@ -340,8 +340,7 @@ with the resulting effective action as a function of $m$ alone given by
 \end{equation}
 This function is plotted as a function of $m$ in Fig.~\ref{fig:action} for a variety of $V_0$ and $f$.
 To finish evaluating the integral, this expression should be maximized with
-respect to $m$. If $m_*$ is such a maximum, then $\overline{\chi(\Omega)}\propto e^{N\mathcal S_\chi(m_*)}$. The order parameter $m$ is both physical and interpretable, as
-it gives the overlap of the configuration $\mathbf x$ with the height axis
+respect to $m$. If $m_*$ is such a maximum, then $\overline{\chi(\Omega)}\propto e^{N\mathcal S_\chi(m_*)}$. The order parameter $m$ is the overlap of the configuration $\mathbf x$ with the height axis
 $\mathbf x_0$. Therefore, the value $m$ that maximizes this action can be
 understood as the latitude on the sphere where most of the contribution to the
 Euler characteristic is made.
@@ -431,7 +430,13 @@ To address this ambiguity, we compute also the average of the square of the Eule
 characteristic, $\overline{\chi(\Omega)^2}$, with details in
 Appendix~\ref{sec:rms}. This has the benefit of always being positive, so that
 the saddle-point approach to the calculation at large $N$ does not produce
-complex values even when $\overline{\chi(\Omega)}$ is negative. We find three
+complex values even when $\overline{\chi(\Omega)}$ is negative. Under the restriction that $f(0)=0$,\footnote{
+  This restriction is equivalent to having no random constant term in the
+  constraint equations. It provides a simplification here because when it is
+  present the replica symmetric (\textsc{rs}) description of this problem can
+  have $q_0>0$, and $\overline{\chi(\Omega)^2}\neq[\overline{\chi(\Omega)}]^2$
+  always.
+}we find three
 saddle points that could contribute to the value of
 $\overline{\chi(\Omega)^2}$: two at $\pm m^*$ where
 $\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$,
@@ -1230,7 +1235,7 @@ and where $\Delta_{12}=G_{12}G_{21}-R_{12}R_{21}$. The expression must be
 extremized over all the order parameters. We look for solutions in two regimes
 that are commensurate with the solutions found for the Euler characteristic.
 These correspond to $m_1=m_2=0$ and $C_{12}=0$, and $m_1=m_2=\pm m_*$ and
-$C_{12}=1$. We find such solutions, and in all cases they have
+$C_{12}=1$. We restrict ourselves to cases with $f(0)=0$, which correspond to constraint equations without a random constant term. We find such solutions, and in all cases they have
 \begin{align}
   G_{12}=G_{21}=R_{12}=R_{21}=D_{12}=\hat m_1=\hat m_2=0
   \\
-- 
cgit v1.2.3-70-g09d2

