1 files changed, 25 insertions, 28 deletions
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 77d1304..3f64012 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -641,7 +641,7 @@ abrupt transition and the Yang--Lee point, the coefficients in the analytic
 part $G$ of the scaling function, and the coefficients in the undetermined
 coordinate function $g$.
 
-The other parameters $B$, $C_0$, $\theta_{YL}$, and $C_{YL}$ are determined by known properties.
+The other parameters $B$, $C_0$, $\theta_{YL}$, and $C_{YL}$ are determined by known properties. {\color{blue} XXX Is this right?}
 For $\theta>\theta_0$, the form \eqref{eq:essential.singularity} can be
 expanded around $\theta=\theta_0$ to yield
 \begin{equation}
@@ -702,28 +702,32 @@ $\theta=0$ and $\theta=\theta_0$, weighted by the uncertainty in the value of
 the known coefficients or by a machine-precision cutoff, whichever is larger.
 We also add the difference between the predictions for $A_\mathrm{YL}$ and
 $\xi_\mathrm{YL}$ and their known numeric values, again weighted by their
-uncertainty. In order to encourage convergence, we also add to the cost the
-weighted coefficients $j!g_j$ and $j!G_j$ defining the function $g$ and $G$ in
-\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery}. This can be
-interpreted as a prior which expects these functions to be analytic, and
-therefore have series coefficients which decay with a factorial.
-
-A Levenberg--Marquardt algorithm is performed on the cost function to find a
-parameter combination which minimizes it. As larger polynomial order in the
+uncertainty. In order to encourage convergence, we also add weak residuals
+$j!g_j$ and $j!G_j$ encouraging the coefficients of the analytic functions $g$ and $G$ in
+\eqref{eq:schofield.funcs} and \eqref{eq:analytic.free.enery} to stay small. {\color{blue} Are these multiplied by a small constant? Should they have an $1/R^n$ for the expected radius of convergence?} This can be interpreted as a prior which expects these functions to be analytic, and therefore have series coefficients which decay with a factorial.
+
+A Levenberg--Marquardt algorithm is performed on the cost function to find a parameter combination which minimizes it. As larger polynomial order in the
 series are fit, the truncations of $G$ and $g$ are extended to higher order so
 that the codimension of the fit is constant.
-
-We performed this procedure starting at $n=2$, or matching the scaling
-function at the low and high temperature zero field points to quadratic order,
+We performed this procedure starting at $n=2$ (matching the scaling
+function at the low and high temperature zero field points to quadratic order), up
 through $n=6$. At higher order we began to have difficulty minimizing the cost.
-The resulting fit coefficients can be found in Table \ref{tab:data} without any
-sort of uncertainty, which is difficult to quantify directly due to the
-truncation of series. However, precise results exist for the value of the
+The resulting fit coefficients can be found in Table \ref{tab:fits}.
+
+Precise results exist for the value of the
 scaling function and its derivatives at the critical isotherm, or equivalently
 for the series coefficients of the scaling function $\mathcal F_0$. Since we do
 not use these coefficients in our fits, the error in the approximate scaling
 functions and their derivatives can be evaluated by comparison to their known
 values at the critical isotherm, or $\theta=1$.
+The difference between the numeric values of the coefficients $\mathcal
+F_0^{(m)}$ and those predicted by the iteratively fit scaling functions are
+shown in Fig.~\ref{fig:error}. For the values for which we were able to make a
+fit, the error in the function and its first several derivatives appear to
+trend exponentially towards zero in the polynomial order $n$. The predictions
+of our fits at the critical isotherm can be compared with the numeric values to
+higher order in Fig.~\ref{fig:phi.series}, where the absolute values of both
+are plotted. {\color{blue} XXX Is the critical isotherm still $\theta=1$ with our general coordinate change?}
 
 \begin{table}
   \begin{tabular}{r|lll}
@@ -890,7 +894,7 @@ values at the critical isotherm, or $\theta=1$.
     Free parameters in the fit of the parametric coordinate transformation and
     scaling form to known values of the scaling function series coefficients
     for $\mathcal F_\pm$. The fit at stage $n$ matches those coefficients up to
-    and including order $n$.
+    and including order $n$. Error estimates are difficult to quantify directly.
   } \label{tab:fits}
 \end{table}
 
@@ -921,18 +925,11 @@ values at the critical isotherm, or $\theta=1$.
     with respect to $\eta$ evaluated at $\eta=0$, as a function of the
     polynomial order $n$ at which the scaling function was fit. The point
     $\eta=0$ corresponds to the critical isotherm at $T=T_c$ and $H>0$, roughly
-    midway between the limits used in the fit at $H=0$ and $T\neq T_c$.
+    midway between the two limits used in the fit, at $H=0$ and $T$ above and below $T_c$. Convergence here should reflect overall convergence of our scaling function at all $\theta$. {\color{blue} Thicker lines? Larger symbols?}
   } \label{fig:error}
 \end{figure}
 
-The difference between the numeric values of the coefficients $\mathcal
-F_0^{(m)}$ and those predicted by the iteratively fit scaling functions are
-shown in Fig.~\ref{fig:error}. For the values for which we were able to make a
-fit, the error in the function and its first several derivatives appear to
-trend exponentially towards zero in the polynomial order $n$. The predictions
-of our fits at the critical isotherm can be compared with the numeric values to
-higher order in Fig.~\ref{fig:phi.series}, where the absolute values of both
-are plotted.
+
 
 \begin{figure}
   \begin{gnuplot}[terminal=epslatex]
@@ -954,7 +951,7 @@ are plotted.
   \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}.
+    \ref{tab:data}. {\color{blue} XXX Numerics is invisible? Why is the decay indicating a radius of convergence larger than $\theta_0$? Mention relation of decay to $\theta_0$? Or even plot $F_0^{(m)} \theta_0^m$ to make points easier to see?}
   } \label{fig:phi.series}
 \end{figure}
 
@@ -992,7 +989,7 @@ Fig.~\ref{fig:phi.series}.
     The series coefficients for the scaling function $\mathcal F_-$ as a
     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}.
+    accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in Caselle's form.
   } \label{fig:glow.series}
 \end{figure}
 
@@ -1020,7 +1017,7 @@ Fig.~\ref{fig:phi.series}.
     The series coefficients for the scaling function $\mathcal F_+$ as a
     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}.
+    accurate scaling function listed in \cite{Caselle_2001_The}. Note all agree well for $H$ near zero, $T > T_c$. {\color{blue} Caselle invisible?}
   } \label{fig:ghigh.series}
 \end{figure}