Merge branch 'aps' of git:research/first_order_singularities/paper into apsaps

1 files changed, 10 insertions, 8 deletions

diff --git a/ising_scaling.tex b/ising_scaling.tex
index a1c8205..5881786 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -181,7 +181,7 @@ use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$.
 The analyticity of the free energy at places away from the critical point
 implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law
 expansions of their arguments about zero, the result of so-called Griffiths
-analyticity. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is
+analyticity \cite{Griffiths_1967_Thermodynamic}. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is
 no phase transition, and the free energy must be an analytic function of its
 arguments. It follows that $\mathcal F_0$ is analytic about zero. This is not
 the case at infinity: since
@@ -245,7 +245,7 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant
 
 
 \begin{figure}
-  \includegraphics{figs/F_lower_singularities}
+  \includegraphics{F_lower_singularities}
   \caption{
     Analytic structure of the low-temperature scaling function $\mathcal F_-$
     in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The circle
@@ -292,7 +292,7 @@ branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant
 $\xi_{\mathrm{YL}}$.
 
 \begin{figure}
-  \includegraphics{figs/F_higher_singularities}
+  \includegraphics{F_higher_singularities}
   \caption{
     Analytic structure of the high-temperature scaling function $\mathcal F_+$
     in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The squares
@@ -421,7 +421,7 @@ $\theta$. Therefore,
 The location $\theta_0$ is not fixed by any principle.
 
 \begin{figure}
-  \includegraphics{figs/F_theta_singularities}
+  \includegraphics{F_theta_singularities}
   \caption{
     Analytic structure of the global scaling function $\mathcal F$ in the
     complex $\theta$ plane. The circles depict essential singularities of the
@@ -478,7 +478,7 @@ Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes
 its real part up to an analytic even function $G(\theta)$, real for real $\theta$.
 
 \begin{figure}
-  \includegraphics{figs/contour_path}
+  \includegraphics{contour_path}
   \caption{
     Integration contour over the global scaling function $\mathcal F$ in the
     complex $\theta$ plane used to produce the dispersion relation. The
@@ -835,7 +835,7 @@ values of both are plotted.
     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 $n$. Error estimates are difficult to quantify directly.
+    and including order $n$. Uncertainty estimates are difficult to quantify directly.
   } \label{tab:fits}
 \end{table}
 
@@ -880,7 +880,7 @@ 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}. The deviation at high polynomial order illustrates the lack of the essential singularity in Caselle's form.
+    accurate scaling function listed in \cite{Caselle_2001_The}. The deviation at high polynomial order illustrates the lack of the essential singularity in the form of Caselle \textit{et al.}.
   } \label{fig:glow.series}
 \end{figure}
 
@@ -951,7 +951,9 @@ the ratio.
     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, which should be zero due to the essential singularity (as seen in the known numeric values and in this work). Cassel {\em et al} do not incorporate the essential singularity.
+    the series, which should be zero due to the essential singularity (as seen
+    in the known numeric values and in this work). Caselle \textit{et al.} do
+    not incorporate the essential singularity.
   } \label{fig:glow.radius}
 \end{figure}