Merge branch 'master' of git.kent-dobias.com:research/first_order_singularities/paper

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', \