From f624a845072b785ca26315463d6f984ff4878382 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 3 Oct 2018 21:52:26 -0400
Subject: added Jim's suggestions

---
 monte-carlo.tex | 101 ++++++++++++++++++++++++++++++++------------------------
 1 file changed, 58 insertions(+), 43 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 578ae14..cf45bcd 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -109,7 +109,7 @@ of the spin--spin coupling in a generic class of spin systems,
 than the introduction of a `ghost spin,\!' our representation relies on
 introducing a `ghost transformation.\!'
 
-\section{Generalized Clusters Without a Field}
+\section{Clusters Without a Field}
 
 We will pose the problem in a general way, but several specific examples can
 be found in Table~\ref{table:models} for concreteness. Let $G=(V,E)$ be a graph, where the set of vertices $V=\{1,\ldots,N\}$
@@ -201,9 +201,11 @@ in the following way.
   \item Pick a random site $m_0$ and add it to the stack.
 
   \item Select a rotation $r\in R_2$ distributed by $f(r\mid m_0,\set s)$.
-    Typically $f$ is uniform on $R_2$, but it is sufficient for preserving
+    Often $f$ is taken as uniform on $R_2$, but it is sufficient for preserving
     detailed balance that $f$ be any function of the seed site $m_0$ and
-    $Z(s,r\cdot s)$ for all $s\in\set s$, which is what we will assume here.
+    $Z(s,r\cdot s)$ for all $s\in\set s$. The flexibility offered by the
+    choice of distribution will be useful in situations where the state space
+    is infinite.
   \item While the stack isn't empty,
     \begin{enumerate}
       \item pop site $m$ from the stack.
@@ -379,7 +381,7 @@ authors in an existing interactive Ising simulator at
 \texttt{https://mattbierbaum.github.io/ising.js}
 \cite{bierbaum_ising.js_nodate}.
 
-\subsection{The $\mathrm O(n)$ model}
+\subsection{The XY and other $\mathrm O(n)$ models}
 \label{sec:examples:on}
 
 In the $\mathrm O(n)$ model spins are described by vectors on the
@@ -422,23 +424,26 @@ limit of large $L$ the expected square angle between neighbors being
 \[
   \avg{\theta^2}\simeq\frac{(n-1)T}{D+H/2}.
 \]
-Fig.~\ref{fig:generator_times} shows the effect of making such a choice on
-autocorrelation times for a critical \threedee $\mathrm O(2)$ model. At small
-fields both methods perform the same as zero field Wolff. Intermediate field
-values see efficiency gains for both methods. At large field the uniform
-sampling method sees correlation times grow rapidly, while for the sampling
-method described here the correlation time crosses over to a constant. A
-similar behavior holds for the critical $\mathrm O(3)$ model, though in that
-case the constant value the correlation time approaches at large field is
-larger than that at zero field (see Fig.~\ref{fig:correlation_time-collapse}).
-More detailed discussion on correlation times and these numeric experiments
-can be found in section \ref{sec:performance}.
+We take $\sigma=\sqrt{\avg{\theta^2}}/2$. Fig.~\ref{fig:generator_times} shows
+the effect of making such a choice on autocorrelation times for a critical
+\threedee \textsc{xy} ($\mathrm O(2)$) model. At small fields both methods
+perform the same as zero field Wolff.  Intermediate field values see
+efficiency gains for both methods. At large field the uniform sampling method
+sees correlation times grow rapidly, while for the sampling method described
+here the correlation time crosses over to a constant. A similar behavior holds
+for the critical $\mathrm O(3)$ model, though in that case the constant value
+the correlation time approaches at large field is larger than its minimum
+value (see Fig.~\ref{fig:correlation_time-collapse}). This behavior isn't
+particularly worrisome, since the very large field regime corresponds to
+correlation lengths smaller than the lattice spacing and is well-described by
+other algorithms.  More detailed discussion on correlation times and these
+numeric experiments can be found in section \ref{sec:performance}.
 
 \begin{figure}
   \include{fig_generator-times}
   \caption{
     The scaled autocorrelation time of the energy $\H$ for the Wolff algorithm
-    on a $32\times32\times32$ O(2) model at its critical temperature as a
+    on a $32\times32\times32$ \textsc{xy} model at its critical temperature as a
     function of applied vector field magnitude $|H|$. Red points correspond to
     reflections sampled uniformly, while the green points represent
     reflections sampled as described in section \ref{sec:examples:on}.
@@ -446,12 +451,7 @@ can be found in section \ref{sec:performance}.
   \label{fig:generator_times}
 \end{figure}
 
-While this approach ameliorates the inefficiency at large field, it is likely
-not the best solution. Here we have set the scale of transformation based on
-the average difference between nearest neighbors, but cluster methods succeed
-because they tend to produce changes on the order of the system's correlation
-length. A more nuanced analysis that samples reflections with producing this
-behavior in mind may perform much better. 
+
 
 \subsection{The Potts model}
 
@@ -530,13 +530,24 @@ one should expect its natural extension in the presence of a field to scale
 roughly like $h^{-z\nu/\beta\delta}$ and collapse appropriately as a function
 of $hL^{\beta\delta/\nu}$.
 
-We measured the autocorrelation time of the energy $\H$ for a variety of
+We measured the autocorrelation time $\tau$ of the energy $\H$ for a variety of
 models at critical temperature with many system sizes and canonical fields
-(see Table~\ref{table:models} with $h=\beta H$) using standard methods
-\cite{ossola_dynamic_2004}. The resulting scaling behavior, plotted in
+(see Table~\ref{table:models} with $h=\beta H$) using standard methods for
+obtaining the value and uncertainty from timeseries
+\cite{ossola_dynamic_2004}. Since the computational effort expended in each
+step of the algorithm depends linearly on the size of the associated cluster,
+these values are then scaled by the average cluster size per site
+$\avg{s_{\text{\sc 1c}}}/L^D$ to produce something proportional to machine
+time. The resulting scaling behavior, plotted in
 Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with an
 extension to finite field of the behavior at zero field, with an eventual
-finite-size crossover to constant autocorrelation time.
+finite-size crossover to constant autocorrelation time. This crossover isn't
+always kind to the efficiency, e.g., in the $\mathrm O(3)$ model, but in the
+large-field regime where the crossover happens the correlation length is on
+the scale of the lattice spacing and better algorithms exist, like
+Bortz--Kalos--Lebowitz for the Ising model \cite{bortz_new_1975}. Also plotted
+are lines proportional to $h^{-z\nu/\beta\delta}$, which match the behavior of
+the correlation times in the intermediate scaling region.
 
 \begin{figure*}
   \include{fig_correlation-times}
@@ -581,21 +592,21 @@ to the scaling functions of the magnetization and susceptibility per site by
   \begin{aligned}
     \avg{s_{\text{\sc 1c}}}
     &=L^{D}\avg{M^2}=\beta\avg\chi+L^{D}\avg{M}^2\\
-    &=L^{\gamma/\nu}\big[(hL^{\beta\delta/\nu},ht^{-\beta\delta})^{-\gamma/\beta\delta}\beta \mathcal
-      Y(hL^{\beta\delta/\nu,ht^{-\beta\delta}})\\
-      &\hspace{1em}+(hL^{\beta\delta/\nu},ht^{-\beta\delta})^{2/\delta}\mathcal
+    &=L^{\gamma/\nu}\big[(hL^{\beta\delta/\nu})^{-\gamma/\beta\delta}\beta \mathcal
+      Y(hL^{\beta\delta/\nu},ht^{-\beta\delta})\\
+      &\hspace{1em}+(hL^{\beta\delta/\nu})^{2/\delta}\mathcal
     M(hL^{\beta\delta/\nu},ht^{-\beta\delta})\big].
   \end{aligned}
 \]
-We therefore expect that, for the Ising model, $\avg{s_{\text{\sc 1c}}}$
-should go as $(hL^{\beta\delta})^{2/\delta}$ for large argument. We further
-conjecture that this scaling behavior should hold for other models whose
-critical points correspond with the percolation transition of Wolff clusters.
-This behavior is supported by our numeric work along the critical isotherm for
-various Ising, Potts, and $\mathrm O(n)$ models, shown in
-Fig.~\ref{fig:cluster_scaling}. Fields for the Potts and $\mathrm O(n)$ models
-take the form $B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$ and
-$B(s)=(h/\beta)[1,0,\ldots,0]s$ respectively. As can be seen, the average
+We therefore expect that, for the Ising model, $\avg{s_{\text{\sc
+1c}}}L^{-\gamma/\nu}$ should go as $(hL^{\beta\delta/\nu})^{2/\delta}$ for
+large argument. We further conjecture that this scaling behavior should hold
+for other models whose critical points correspond with the percolation
+transition of Wolff clusters.  This behavior is supported by our numeric work
+along the critical isotherm for various Ising, Potts, and $\mathrm O(n)$
+models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the Potts and
+$\mathrm O(n)$ models take the form $B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$
+and $B(s)=(h/\beta)[1,0,\ldots,0]s$ respectively. As can be seen, the average
 cluster size collapses for each model according to the scaling hypothesis, and
 the large-field behavior likewise scales as we expect from the na\"ive Ising
 conjecture.
@@ -614,8 +625,8 @@ conjecture.
 
 \section{Applying Nonlinear Fields to the xy Model}
 
-This far our numeric work has quantified the performance of existing
-techniques. Here, we apply our general framework in a new way:
+Thus far our numeric work has quantified the performance of existing
+techniques. Briefly, we demonstrate our general framework in a new way:
 harmonic perturbations to the low-temperature {\sc xy}, or \twodee O(2),
 model. We consider fields of the form $B_n(s)=h_n\cos(n\theta(s))$, where
 $\theta$ is the angle made between $s$ and the $x$-axis, say. Corrections of
@@ -634,7 +645,7 @@ sufficiently high temperatures below the Kosterlitz--Thouless point
 \begin{figure}
   \include{fig_harmonic-susceptibilities}
   \caption{Susceptibilities as a function of system size for a \twodee O(2)
-    model at $T=0.7$ and with (top) fourfold symmetric and (bottom) eightfold
+    model at $T=0.7$ and with (top) fourfold symmetric and (bottom) sixfold
     symmetric perturbing fields. Different field strengths are shown in
     different colors.
   }
@@ -644,7 +655,7 @@ sufficiently high temperatures below the Kosterlitz--Thouless point
 We made a basic investigation of this result using our algorithm. Since we ran
 the algorithm at fairly high fields we did not choose reflections though the
 origin uniformly. Instead, we choose the planes of reflection first by
-rotating our starting spin by $2\pi m/n$ degrees for $m$ uniformly taken from
+rotating our starting spin by $\pi m/n$ for $m$ uniformly taken from
 $1,\ldots,n$ and generating a normal to the plane from that direction as
 described in Section \ref{sec:examples:on}. The resulting susceptibilities as
 a function of system size are shown for various field strengths in
@@ -652,7 +663,11 @@ Fig.~\ref{fig:harmonic-susceptibilities}. In the fourfold case, for each field
 strength there is a system size at which the divergence in the susceptibility
 is cut off, while for the sixfold case we measured no such cutoff, even up to
 strong fields. This conforms to the expected result, that even in a strong field
-the sixfold perturbations preserve the critical behavior.
+the sixfold perturbations preserve the critical behavior. Previous work has
+used Monte Carlo to investigate similar symmetry-breaking fields and used a hybrid
+cluster--metropolis method \cite{ala-nissila_numerical_1994}. To our
+knowledge, no application of a direct cluster method has been applied to this
+problem before now.
 
 \section{Conclusions}
 
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