revisions from Jim

1 files changed, 32 insertions, 21 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 9b61ba1..584dba9 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -54,14 +54,16 @@
 \begin{abstract}
   We introduce a natural way to extend celebrated spin-cluster Monte Carlo
   algorithms for fast thermal lattice simulations at criticality, like Wolff,
-  to systems in arbitrary fields. By generalizing the `ghost spin'
-  representation to one with a `ghost transformation,\!' global invariance to
-  spin symmetry transformations is restored at the cost of an extra degree of
-  freedom. The ordinary cluster-building process can then be run on the new
-  representation. We show that this extension preserves the scaling of
-  accelerated dynamics in the absence of a field for several canonical systems
-  and demonstrate the method's use in modelling the presence of novel
-  nonlinear fields.
+  to systems in arbitrary fields, be they linear magnetic vector fields or
+  nonlinear anisotropic ones. By generalizing the `ghost spin' representation
+  to one with a `ghost transformation,\!' global invariance to spin symmetry
+  transformations is restored at the cost of an extra degree of freedom which
+  lives in the space of symmetry transformations. The ordinary
+  cluster-building process can then be run on the new representation. We show
+  that this extension preserves the scaling of accelerated dynamics in the
+  absence of a field for Ising, Potts, and $\mathrm O(n)$ models and
+  demonstrate the method's use in modelling the presence of novel nonlinear
+  fields.
 \end{abstract}
 
 \maketitle
@@ -143,13 +145,15 @@ The Hamiltonian of this system is a function $\H:X^N\to\R$ defined by
 where $\J:X\times X\to\R$ couples adjacent spins and $B:X\to\R$ is an external
 field. $\J$ must be symmetric in its arguments and invariant under the action
 of any element of $R$ applied to the entire lattice, that is, for any $r\in R$
-and $s,t\in X$, $\J(r\cdot s,r\cdot t)=\J(s,t)$.  One may also allow $\J$ to also
-be a function of edge---for modelling random-bond, long-range, or anisotropic
-interactions---or allow $B$ to be a function of site---for applying arbitrary
-boundary conditions or modelling random fields. The formal results of this
-paper (that the algorithm obeys detailed balance and ergodicity) hold equally
-well for these cases, but we will drop the additional index notation for
-clarity. Statements about efficiency may not hold.
+and $s,t\in X$, $\J(r\cdot s,r\cdot t)=\J(s,t)$.  One may also allow $\J$ to
+also be a function of edge---for modelling random-bond, long-range, or
+anisotropic interactions---or allow $B$ to be a function of site---for
+applying arbitrary boundary conditions or modelling random fields. The formal
+results of this paper (that the algorithm obeys detailed balance and
+ergodicity) hold equally well for these cases, but we will drop the additional
+index notation for clarity. Some extensions, like adding strong random fields
+or bonds, ultimately prove inefficient \cite{rieger_monte_1995,
+redner_graphical_1998}.
 
 \begin{table*}[htpb]
   \begin{tabular}{l||ccccc}
@@ -380,7 +384,8 @@ in the cluster for the $\mathrm O(n)$), but really what it estimates is the
 averaged squared magnetization, which corresponds to the susceptibility when
 the average magnetization is zero. At finite field the latter thing is no
 longer true, but the correspondence between cluster size (sum of
-$x$-components) and the squared magnetization continues to hold.
+$x$-components) and the squared magnetization continues to hold (see
+\eqref{eq:cluster-size-scaling} and Fig.~\ref{fig:cluster_scaling} below).
 
 \section{Examples}
 \label{sec:examples}
@@ -460,8 +465,8 @@ 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
+corresponds to correlation lengths comparable to the lattice spacing and is
+efficiently simulated by other algorithms.  More detailed discussion on correlation
 times and these numeric experiments can be found in section
 \ref{sec:performance}.
 
@@ -538,7 +543,7 @@ algorithm into that domain. Some systems are not efficient under Wolff, and we
 don't expect them to fare better when extended in a field. For instance, Ising
 models with random fields or bonds technically can be treated with Wolff
 \cite{dotsenko_cluster_1991}, but it is not efficient because the clusters
-formed do scale naturally with the correlation length \cite{rieger_monte_1995,
+formed do not scale naturally with the correlation length \cite{rieger_monte_1995,
 redner_graphical_1998}. Other approaches, like replica methods, should be
 relied on instead \cite{redner_graphical_1998, chayes_graphical_1998,
 machta_replica-exchange_2000}. 
@@ -569,7 +574,9 @@ of $hL^{\beta\delta/\nu}$.
     \twodee 3-State Potts: 0.55(1), \twodee 4-State Potts: 0.94(5),
     \threedee O(2): 0.17(2), \threedee O(3): 0.13(2). $\mathrm O(n)$ models
     use the distribution of transformations described in Section
-    \ref{sec:examples:on}.
+    \ref{sec:examples:on}. The curves stop collapsing at high fields when the
+    correlation length falls to near the lattice spacing; here non-cluster
+    algorithms can be efficiency used.
   }
   \label{fig:correlation_time-collapse}
 \end{figure*}
@@ -621,6 +628,7 @@ proportional to their size, or
     1c}}(s)=\sum_ss\frac sNP_{\text{SW}}(s)\\
     &=L^{\gamma/\nu}g(ht^{-\beta\delta},hL^{\beta\delta/\nu}).
   \end{aligned}
+  \label{eq:cluster-size-scaling}
 \]
 
 \begin{figure*}
@@ -678,7 +686,10 @@ ala-nissila_numerical_1994, dierker_consequences_1986, selinger_theory_1988}
 results that emerge from systems with one or more of these fields applied, it
 is predicted that $h_4$ is relevant while $h_6$ is not at some sufficiently
 high temperatures below the Kosterlitz--Thouless point
-\cite{jose_renormalization_1977}.
+\cite{jose_renormalization_1977}. The sixfold fields are expected to be
+present, for instance, in the otherwise Kosterlitz--Thouless-type
+two-dimensional melting of argon on a graphite substrate
+\cite{zhang_melting_1991}.
 
 \begin{figure}
   \include{fig_harmonic-susceptibilities}