From 03de8ceab441fddaba727f2bcd362275f6ced11e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 30 Jun 2022 20:48:29 +0200
Subject: More interpretation work.

---
 frsb_kac-rice.tex | 20 ++++++++++++++------
 1 file changed, 14 insertions(+), 6 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 130738d..a01d803 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -315,8 +315,7 @@ We introduce new fields
   R_{ab}=-i\frac1N\hat s_a\cdot s_b &&
   D_{ab}=\frac1N\hat s_a\cdot\hat s_b
 \end{align}
-$C_{ab}$ is the overlap between spins belonging to different replicas.
-
+Their physical meaning is explained in \S\ref{sec:interpretation}.
 By substituting these parameters into the expressions above and then making a
 change of variables in the integration from $s_a$ and $\hat s_a$ to these three
 matrices, we arrive at the form for the complexity
@@ -636,6 +635,7 @@ Here a picture of $\chi$ vs $C$ or $X$ vs $C$ showing limits $q_{max}$, $x_{max}
 for different energies and typical vs minima.
 
 \section{Interpretation}
+\label{sec:interpretation}
 
 Let $\langle A\rangle$ be the average of $A$ over stationary points with given $E$ and $\mu$, i.e.,
 \begin{equation}
@@ -649,20 +649,28 @@ with
   d\nu(s)=ds\,\delta(NE-H(s))\delta(\partial H(s)+\mu s)|\det(\partial\partial H(s)+\mu I)|
 \end{equation}
 the Kac--Rice measure. The fields $C$, $R$, and $D$ defined in
-\eqref{eq:fields} can be related to certain averages of this type. First,
+\eqref{eq:fields} can be related to certain averages of this type.
+
+\subsection{\textit{C}: distribution of overlaps}
+
+First,
 consider $C$, which has an interpretation nearly identical to that of Parisi's
 $Q$ matrix of overlaps. It can be shown that its off-diagonal corresponds to
 the probability distribution of the overlaps between stationary points $P(q)$. First, define this distribution as
 \begin{equation}
   P(q)=\frac1{\mathcal N^2}\sum_{\sigma,\sigma'}\delta\left(\frac{s_\sigma\cdot s_{\sigma'}}N-q\right)
 \end{equation}
-It is straightforward to show that moments of this distribution are related to certain averages of the form
+where the sum is twice over stationary points $\sigma$ and $\sigma'$.  It is
+straightforward to show that moments of this distribution are related to
+certain averages of the form
 \begin{equation}
   \int dq\,q^p P(q)
   =q^{(p)}
   \equiv\frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle
 \end{equation}
-In particular, the appeal of Parisi to properties of pure states is unnecessary here, since the stationary points are points. These moments are related to our $C$ by computing their average over disorder:
+The appeal of Parisi to properties of pure states is unnecessary here, since
+the stationary points are points. These moments are related to our $C$ by
+computing their average over disorder:
 \begin{equation}
   \begin{aligned}
     \overline{q^{(p)}}
@@ -676,7 +684,7 @@ where $C$ is assumed to take its saddle point value. If we now change variables
 \begin{equation}
   \overline{q^{(p)}}=\int dc\,c^p\frac{dx}{dc}
 \end{equation}
-from which we conclude $\overline{P(q)}=\frac{dx}{dc}|_{c=q}$.
+from which we conclude $\overline{P(q)}=\frac{dx}{dc}\big|_{c=q}$.
 
 With this
 established, we now address what it means for $C$ to have a nontrivial
-- 
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