From 32a3ee96ae4ae5a54b82365491a40edfc2212494 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 23 Oct 2021 22:46:11 +0200
Subject: Writing

---
 ising_scaling.tex | 28 +++++++++++++++++-----------
 1 file changed, 17 insertions(+), 11 deletions(-)

diff --git a/ising_scaling.tex b/ising_scaling.tex
index c7e7adb..04cc0eb 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -137,16 +137,21 @@ to constant rescaling of $u_h$). The invariant scaling combinations that appear
 as the arguments to the universal scaling functions will come up often, and we
 will 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. 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 $\mathcal F_0(\eta)=\eta^{D\nu}\mathcal
-F_\pm(\eta^{-1/\Delta})$ has a power-law expansion about zero, $\mathcal
-F_\pm(\xi)\sim \xi^{D\nu/\Delta}$ for large $\xi$. The nonanalyticity of
-these functions at infinite argument can therefore be understood as an artifact
-of the chosen coordinates.
+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
+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
+\begin{equation}
+  \mathcal F_\pm(\xi)
+  =\xi^{D\nu/\Delta}\mathcal F_0(\pm \xi^{-1/\Delta})+\frac1{8\pi}\log\xi^{2/\Delta}
+\end{equation}
+and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal
+F_\pm$ has a series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at
+large $\xi$, along with logarithms. The nonanalyticity of these functions at
+infinite argument can be understood as an artifact of the chosen coordinates.
 
 For the scale of $u_t$ and $u_h$, we adopt the same convention as used by
 \cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on
@@ -210,7 +215,8 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant.
 To lowest order, this singularity is a function of the scaling invariant $\xi$
 alone. It is therefore suggestive that this should be considered a part of the
 singular free energy and moreover part of the scaling function that composes
-it. We will therefore make the ansatz that
+it. There is substantial numeric evidence for this as well. We will therefore
+make the ansatz that
 \begin{equation} \label{eq:essential.singularity}
   \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right]
 \end{equation}
-- 
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