From 90e8393b4069ace54c73975e3159aaf0db782ab7 Mon Sep 17 00:00:00 2001
From: jps6 <jps6@cornell.edu>
Date: Thu, 25 May 2023 15:35:15 +0000
Subject: Update on Overleaf.

---
 referee_response.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

(limited to 'referee_response.tex')

diff --git a/referee_response.tex b/referee_response.tex
index 01ece7a..b33d3b1 100644
--- a/referee_response.tex
+++ b/referee_response.tex
@@ -183,10 +183,11 @@ nothing about this natural boundary which is a major feature of the
 analyticity of the model that must be explained.
 \end{verbatim}
 
-The natural boundary mentioned is purported to exist in the complex temperature
-dependence susceptibility of the lattice Ising model. It is not clear to us
-why it should be present in the scaling function of the free energy for the
-Ising universality class. We have removed the inaccurate comments.
+Our scaling function indeed does not show any evidence of a natural boundary or logarithmic corrections at complex temperatures in a field: we see only the branch cut of the dominant logarithmic singularity in the free energy. This is to be expected, because our calculation focuses on the universal scaling function as it depends upon the relevant variables $t$ and $h$, and does not incorporate singular corrections to scaling from irrelevant operators.
+
+The logarithmic corrections seen in the susceptibility are thought by these authors to come from singular corrections to scaling from these irrelevant operators. Furthermore, these logarithms are thought by Perk (private communication) to be associated with the lattice models, so they should not be seen in (say) the $\phi^4$ theory or membrane Ising phase transitions. 
+
+We expect that a natural boundary in the susceptibility in the complex plane in the lattice model is due to these corrections to scaling, and thus should not be expected to manifest itself in the universal scaling function we calculate.
 
 \begin{verbatim}
 3. There are completely unsubstantiated claims made at the end of the
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