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authorJaron Kent-Dobias <jaron@kent-dobias.com>2021-10-26 12:27:07 +0200
committerJaron Kent-Dobias <jaron@kent-dobias.com>2021-10-26 12:27:07 +0200
commit89f1c347e64a760a9630306da0f839ce5619e96b (patch)
tree83729a51aefba2b3553647fc773156cf1259abc5
parent2242fbad78679e941037b4a319508a4d2ed5a58c (diff)
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Added labels for comparison figures and wrote about them.
-rw-r--r--ising_scaling.tex83
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}