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-rw-r--r--data/phi_comparison.dat14
-rw-r--r--data/yl_comparison.dat14
-rw-r--r--ising_scaling.tex37
3 files changed, 34 insertions, 31 deletions
diff --git a/data/phi_comparison.dat b/data/phi_comparison.dat
index d58e96c..14c1885 100644
--- a/data/phi_comparison.dat
+++ b/data/phi_comparison.dat
@@ -1,8 +1,6 @@
-2 0.002860955318525926 0.004496459219585747 0.0025781014469987568 0.0004361990091461404 0.0015990766131468608
-3 0.0005720429508622171 0.0010847239134089692 0.000805365486839224 0.00006427186448818359 0.0003917599113948043
-4 0.00003961608489011503 0.0001278039316774393 0.00018174532718064074 0.00013408467605927413 0.00002358497641938859
-5 0.0000622987443403833 0.00016919055775577174 0.0002085264051783775 0.0001300161350704411 3.732946060521912e-6
-6 0.00005016392362722222 0.00014150435296594877 0.00016732830408854038 0.00007562595035311148 0.00005202003334785774
-7 0.000015703311988302104 0.00005762691693961264 0.00009304388663239349 0.00007579125219901034 0.000010561158237231718
-8 8.173890766238756e-6 0.000021607891761421527 0.000022883111337773654 6.9772437447900015e-6 0.000010861362342928695
-9 2.4873158455118727e-6 7.768088409076945e-6 9.816343265717231e-6 3.88602466879287e-6 5.648983756909563e-6 \ No newline at end of file
+2 0.0015185988806263317 0.0016424512339846542 0.00010931409748554666 0.0014019918834638455 0.0009799159023033766
+3 0.0007353808953864949 0.0014547698806950948 0.0011618922642676333 0.0001926030528925822 0.0004716780917582721
+4 0.0006832169448554026 0.0015493096608610868 0.001202074914739018 0.00024563197897510894 0.0011902522778089069
+5 0.00003083151013294483 0.000024777126610253664 0.00004067889303242811 0.00008916575299173363 0.00006668594729061653
+6 0.000012555575489514581 0.000011541389969949023 0.00003418912424536791 0.00009040132752613159 0.00009049768674665578
+7 3.513158584711462e-6 5.987223218928417e-6 4.50941314238118e-6 0.000026308155365337843 0.000039381306641833976 \ No newline at end of file
diff --git a/data/yl_comparison.dat b/data/yl_comparison.dat
index f1ece63..3001c98 100644
--- a/data/yl_comparison.dat
+++ b/data/yl_comparison.dat
@@ -1,8 +1,6 @@
-2 0.0156377454168716 0.00005
-3 0.0026092903452304694 0.00005
-4 0.00287653656882067 0.00005
-5 0.0030257297041298703 0.00005
-6 0.0006769621913844392 0.00005
-7 0.0005281357188132441 0.00005
-8 0.0000685310321167365 0.00005
-9 0.000035091061848013805 0.00005 \ No newline at end of file
+2 2.5824766344828554e-7 0.00005
+3 7.646795701476972e-6 0.00005
+4 5.935120440947461e-7 0.00005
+5 0.0013932555059845142 0.00005
+6 0.0005014284531485447 0.00005
+7 0.0002780661068152446 0.00005 \ No newline at end of file
diff --git a/ising_scaling.tex b/ising_scaling.tex
index 32085ab..6b89cb2 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -142,16 +142,21 @@ 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
@@ -215,7 +220,8 @@ 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}
@@ -281,7 +287,7 @@ functions with different asymptotic expansions in different limits, we adopt
another coordinate system, in terms of which a scaling function can be defined
that has polynomial expansions in \emph{all} limits.
-In all dimensions, the Schofield coordinates $R$ and $\theta$ will be implicitly defined by
+The Schofield coordinates $R$ and $\theta$ are implicitly defined by
\begin{align} \label{eq:schofield}
u_t(R, \theta) = R(1-\theta^2)
&&
@@ -700,13 +706,14 @@ accuracy of the fit results can be checked against the known values here.
dat = 'data/phi_comparison.dat'
set xlabel '$n$'
- set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$'
+ set xrange [1.5:7.5]
+ set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$'
set format y '$10^{%T}$'
- set style data linespoints
set logscale y
+
+ set style data linespoints
set key title '\raisebox{0.5em}{$m$}' bottom left
- set xrange [1.5:9.5]
plot \
dat using 1:2 title '0', \