From 91be15cd9d336cdd96c7bc8468c3a6f62d0ebba6 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 10 Mar 2025 11:40:19 -0300
Subject: Change addressing report #1, second comment/typo
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Added a footnote discussing the commutation of the limits N→∞ and α→0 or
M=1.
---
 topology.tex | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 281831c..11f1277 100644
--- a/topology.tex
+++ b/topology.tex
@@ -634,7 +634,9 @@ into the structure of solutions in this regime is merited.
 When $M=1$ the solution manifold corresponds to the energy
 level set of a spherical spin glass with energy density $E=V_0/\sqrt N$. All the
 results from the previous sections follow, and can be translated to the spin
-glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed. With a little algebra this procedure yields
+glasses by taking the limit $\alpha\to0$ while keeping $E=V_0\alpha^{1/2}$ fixed.\footnote{
+  It is plausible that the limits of $N\to\infty$ implicit in the saddle point expansion and the limit of $\alpha\to0$ taken here do not commute, and that $M=1$ should be taken from the beginning of the calculation. However, in this case the two procedures do commute. The $\alpha\to0$ limit accomplishes only the elimination of the first term from the effective action \eqref{eq:S.m}, while following Appendix~\ref{sec:euler} with $M=1$ from the outset results in the same term not appearing in the effective action because it is of subleading order in $N$.
+} With a little algebra this procedure yields
 \begin{align}
   E_\text{on}=\pm\sqrt{2f(1)}
   &&
-- 
cgit v1.2.3-70-g09d2

