From 22220b4b413822eaec94b2896192d5c8cf9b32d2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jun 2024 11:54:29 +0200
Subject: Editing changes to the superspace subspection and a figure caption.

---
 marginal.tex | 17 ++++++++++++-----
 1 file changed, 12 insertions(+), 5 deletions(-)

diff --git a/marginal.tex b/marginal.tex
index 190c3be..3eeed6b 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -650,9 +650,9 @@ The ordinary Kac--Rice calculation involves many moving parts, and this method
 for incorporating marginality adds even more. It is therefore convenient to
 introduce compact and simplifying notation through a superspace representation.
 The use of superspace in the Kac--Rice calculation is well established, as well
-as the deep connections with BRST symmetry that is implied.
+as the deep connections with BRST symmetry that is implied \cite{Annibale_2003_Supersymmetric, Annibale_2003_The, Annibale_2004_Coexistence}.
 Appendix~\ref{sec:superspace} introduces the notation and methods of
-superspace. Here we describe how it can be used to simplify the complexity
+superspace algebra. Here we describe how it can be used to simplify the complexity
 calculation for marginal minima.
 
 We consider the $\mathbb R^{N|4}$ superspace whose Grassmann indices are
@@ -687,7 +687,13 @@ Here we have also defined the operator
         (1-\hat\beta\bar\theta_1\theta_1)
         -\delta^{\alpha1}\hat\lambda-\beta
 \end{equation}
-which encodes various aspects of the complexity problem, and the measures
+which encodes various aspects of the complexity problem. When the Lagrangian is
+expanded in a series with respect to the Grassmann indices and the definition
+of $B$ inserted, the result of the Grassmann integrals produces exactly the
+content of the integrand in \eqref{eq:min.complexity.expanded} with the
+substitutions \eqref{eq:delta.grad}, \eqref{eq:delta.energy},
+\eqref{eq:delta.eigen}, and \eqref{eq:determinant} of the Dirac $\delta$
+functions and the determinant made. The new measures
 \begin{align}
   d\pmb\phi_a^\alpha
   &=\left[
@@ -702,7 +708,7 @@ which encodes various aspects of the complexity problem, and the measures
   d\pmb\omega&=\bigg(\prod_{i=1}^rd\omega_i\bigg)
   \,\delta\bigg(N\mu-\sum_i^r\omega_i\operatorname{Tr}\partial\partial g_i\bigg)
 \end{align}
-that collect the individual measures of the various fields embedded in the superfield.
+collect the individual measures of the various fields embedded in the superfield, along with their constraints.
 \end{widetext}
 With this way of writing the replicated count, the problem of marginal
 complexity temporarily takes the schematic form of an equilibrium calculation
@@ -710,6 +716,7 @@ with configurations $\pmb\phi$, inverse temperature $B$, and energy $L$. This
 makes the intermediate pieces of the calculation dramatically simpler. Of
 course the intricacies of the underlying problem are not banished: near the end
 of the calculation, terms involving the superspace must be expanded.
+We will make use of this representation to simplify the analysis of the marginal complexity when analyzing random sums of squares in Section \ref{sec:least.squares}.
 
 \section{Examples}
 \label{sec:examples}
@@ -1251,7 +1258,7 @@ subspace and a stable minimum on the other.
     \textbf{(b)}~Spectra corresponding to the parameters $\omega_1$ and
     $\omega_2$ marked by the circles in panel (a).
     \textbf{(c)}~The complexity of marginal minima in a multispherical model with
-    $f_i(q)=q^{p_i}/[p_i(p_i-1)]$ for $p_1=3$ and $p_2=4$ for a variety of
+    $f_1(q)=\frac16q^3$ and $f_2(q)=\frac1{12}q^4$ for a variety of
     $\epsilon$. Since $f_1''(1)=f_2''(1)=1$, the marginal values correspond
     precisely to those in (a--b).
   } \label{fig:msg.marg}
-- 
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