Added labels for comparison figures and wrote about them.

1 files changed, 51 insertions, 32 deletions

diff --git a/ising_scaling.tex b/ising_scaling.tex
index 5c1612d..668ba63 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -797,7 +797,13 @@ error in the function and its first several derivatives appears to trend
 towards zero exponentially in the polynomial order $n$.
 
 Even at $n=2$, where only six unknown parameters have been fit, the results are
-accurate to within $2\times10^{-3}$. This approximation for the scaling functions also captures the singularities at the high- and low-temperature zero-field points well. 
+accurate to within $2\times10^{-3}$. This approximation for the scaling
+functions also captures the singularities at the high- and low-temperature
+zero-field points well. A direct comparison between the magnitudes of the
+series coefficients known numerically and those given by the approximate
+functions is shown for $\mathcal F_-$ in Fig.~\ref{fig:glow.series}, for
+$\mathcal F_+$ in Fig.~\ref{fig:ghigh.series}, and for $\mathcal F_0$ in
+Fig.~\ref{fig:phi.series}.
 
 \begin{figure}
   \begin{gnuplot}[terminal=epslatex]
@@ -830,35 +836,6 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function
 
 \begin{figure}
   \begin{gnuplot}[terminal=epslatex]
-    dat1 = 'data/glow_numeric.dat'
-    dat2 = 'data/glow_series_ours_0.dat'
-    dat3 = 'data/glow_series_ours_7.dat'
-    dat4 = 'data/glow_series_caselle.dat'
-    ratLast(x) = (back2 = back1, back1 = x, back1 / back2)
-    back1 = 0
-    back2 = 0
-
-    set xlabel '$1/m$'
-    set xrange [0:0.55]
-    set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^{(m-1)}$'
-    set yrange [0:15]
-
-    plot \
-      dat1 using (1/$1):(abs(ratLast($2))) title 'Numeric', \
-      dat2 using (1/$1):(abs(ratLast($2))) title 'This work ($n=2$)', \
-      dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=7$)', \
-      dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}'
-  \end{gnuplot}
-  \caption{
-    Sequential ratios of the series coefficients of the scaling function
-    $\mathcal F_-$ as a function of inverse polynomial order $m$. The
-    extrapolated $y$-intercept of this plot gives the radius of convergence of
-    the series.
-  }
-\end{figure}
-
-\begin{figure}
-  \begin{gnuplot}[terminal=epslatex]
     dat1 = 'data/ghigh_numeric.dat'
     dat2 = 'data/ghigh_series_ours_2.dat'
     dat3 = 'data/ghigh_series_ours_7.dat'
@@ -882,7 +859,7 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function
     function of polynomial order $m$. The numeric values are from Table
     \ref{tab:data}, and those of Caselle \textit{et al.} are from the most
     accurate scaling function listed in \cite{Caselle_2001_The}.
-  }
+  } \label{fig:ghigh.series}
 \end{figure}
 
 \begin{figure}
@@ -903,7 +880,49 @@ accurate to within $2\times10^{-3}$. This approximation for the scaling function
       dat3 using 1:(abs($2)) title 'This work ($n=7$)'
   \end{gnuplot}
   \caption{
-  }
+    The series coefficients for the scaling function $\mathcal F_0$ as a
+    function of polynomial order $m$. The numeric values are from Table
+    \ref{tab:data}.
+  } \label{fig:phi.series}
+\end{figure}
+
+Besides reproducing the high derivatives in the series well, the approximate
+functions defined here feature the appropriate singularity at the abrupt
+transition. Fig.~\ref{fig:glow.radius} shows the ratio of subsequent series
+coefficients for $\mathcal F_-$ as a function of the inverse order, which
+should converge in the limit of $m\to0$ to the inverse radius of convergence
+for the series. Approximations for the function without the explicit
+singularity have a nonzero radius of convergence, where both the numeric data
+and the approximate functions defined here show the appropriate divergence in
+the ratio.
+
+\begin{figure}
+  \begin{gnuplot}[terminal=epslatex]
+    dat1 = 'data/glow_numeric.dat'
+    dat2 = 'data/glow_series_ours_0.dat'
+    dat3 = 'data/glow_series_ours_7.dat'
+    dat4 = 'data/glow_series_caselle.dat'
+    ratLast(x) = (back2 = back1, back1 = x, back1 / back2)
+    back1 = 0
+    back2 = 0
+
+    set xlabel '$1/m$'
+    set xrange [0:0.55]
+    set ylabel '$\mathcal F_-^{(m)}/\mathcal F_-^{(m-1)}$'
+    set yrange [0:15]
+
+    plot \
+      dat1 using (1/$1):(abs(ratLast($2))) title 'Numeric', \
+      dat2 using (1/$1):(abs(ratLast($2))) title 'This work ($n=2$)', \
+      dat3 using (1/$1):(abs(ratLast($2))) title 'This work ($n=7$)', \
+      dat4 using (1/$1):(abs(ratLast($2))) title 'Caselle \textit{et al.}'
+  \end{gnuplot}
+  \caption{
+    Sequential ratios of the series coefficients of the scaling function
+    $\mathcal F_-$ as a function of inverse polynomial order $m$. The
+    extrapolated $y$-intercept of this plot gives the radius of convergence of
+    the series.
+  } \label{fig:glow.radius}
 \end{figure}
 
 \section{Outlook}