From b96fb40f105e8d23338d274ff12f293c604a6bf5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 2 Oct 2017 15:52:32 -0400
Subject: final edits before arxiv reupload

---
 essential-ising.tex   | 20 +++++++++++++-------
 figs/fig-series.gplot | 12 ++++++------
 2 files changed, 19 insertions(+), 13 deletions(-)

diff --git a/essential-ising.tex b/essential-ising.tex
index dd2c083..f789eb4 100644
--- a/essential-ising.tex
+++ b/essential-ising.tex
@@ -350,7 +350,7 @@ In forthcoming work,
 we develop a method to incorporate the essential singularity in the scaling
 functions into a form that also incorporates known properties of the scaling
 functions in the rest of the configuration space using a Schofield-like
-parameterization \cite{kent-dobias.2018.parametric}. Briefly, we define
+parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define
 parameters $R$ and $\theta$ by
 \begin{align}
   t=R(1-\theta^2)
@@ -381,7 +381,7 @@ arbitrary analytic additive function $Y$,
   &&
   Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n}
 \end{align}
-so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating thees
+so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these
 coefficients, we can attempt to give the resulting scaling form a series
 expansion consistent with known values. One such prediction---made by fixing
 the first four terms in the low-temperature, critical isotherm, and
@@ -390,6 +390,9 @@ of $\tilde\fX$---is shown as a dashed yellow line in
 Fig.~\ref{fig:scaling_fits}.
 As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values
 exactly up to $n=5$, and improve the agreement with higher-order coefficients.
+Unlike scaling forms which treat $\fX$ as analytic at the coexistence line,
+the series coefficients of the scaling form developed here increase without
+bound at high order.
 
 
 \begin{figure}
@@ -403,10 +406,11 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients.
     and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines
     show our analytic results \eqref{eq:sus_scaling} and
     \eqref{eq:mag_scaling}, the dashed yellow lines show 
-    a scaling function modified to match known series expansions
-    in several known limits, and the
+    a scaling function modified to match known series expansions of the
+    susceptibility
+    to third order, and the
     dotted green lines show the
-    polynomial resulting from truncating the series after the eight terms
+    polynomial resulting from truncating the known series expansion after the eight terms
     reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}.
   }
   \label{fig:scaling_fits}
@@ -418,10 +422,12 @@ exactly up to $n=5$, and improve the agreement with higher-order coefficients.
     The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue
     pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the
     yellow saltires correspond to a scaling function modified to match known
-    series expansions in several known limits, and the green
+    series expansions of the susceptibility to third order---and therefore
+    the free energy to fifth order---and the green
     stars
     correspond to the first eight coefficients from
-    \cite{mangazeev.2008.variational,mangazeev.2010.scaling}.
+    \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified
+    scaling function and the known coefficients match exactly up to $n=5$.
   }
   \label{fig:series}
 \end{figure}
diff --git a/figs/fig-series.gplot b/figs/fig-series.gplot
index 729ff73..49aa4cc 100644
--- a/figs/fig-series.gplot
+++ b/figs/fig-series.gplot
@@ -1,5 +1,5 @@
 
-set terminal pslatex rotate size 3.417,2.111
+set terminal pslatex rotate size 3.6,2.111
 
 cc1 = "#5e81b5"
 cc2 = "#e19c24"
@@ -10,18 +10,18 @@ set logscale y
 
 data = "figs/fig-series-data.dat"
 
-set xrange [0.5:8.5]
+set xrange [0.5:15.5]
 set yrange [0.000005:5]
 set key off
 set xlabel '$n$'
-set ylabel offset 1 '$|f_n|$'
+set ylabel offset 2 '$|f_n|$'
 
 set ytics format ''
 
 set ytics add ('$\footnotesize10^{-5}$' 10**(-5),'$\footnotesize10^{-4}$' 10**(-4), '$\footnotesize10^{-3}$' 10**(-3),'$\footnotesize10^{-2}$' 10**(-2), '$\footnotesize10^{-1}$' 10**(-1),'$\footnotesize10^{0}$' 10**(0))
 
 plot \
-  data using 1:2 with points lc rgb cc1, \
-  data using 1:3 with points lc rgb cc2, \
-  data using 1:4 with points lc rgb cc3
+  data using 1:3 with points lc rgb cc1, \
+  data using 1:2 with points lc rgb cc2, \
+  data using 1:5 with points lc rgb cc3
 
-- 
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