From 0ce1a87356ec6f1035f67a6810313a7388b5f717 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 25 Sep 2018 17:01:19 -0400
Subject: broke up introduction and renamed the first section

---
 monte-carlo.tex | 47 ++++++++++++++++++++++++++---------------------
 1 file changed, 26 insertions(+), 21 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 70cc952..7f1e042 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -72,17 +72,20 @@ distribution of systems states. These Monte Carlo algorithms are better the
 faster they arrive at a statistically independent sample. This typically
 becomes a problem near critical points, where critical slowing down
 \cite{wolff_critical_1990} results in power-law divergences of dynamic
-timescales. Celebrated cluster algorithms largely addressed this in the
-absence of symmetry-breaking fields by using nonlocal updates
-\cite{janke_nonlocal_1998} whose clusters undergo a percolation transition at
-the critical point of the system \cite{coniglio_clusters_1980}.  These result
-in relatively small dynamic exponents for many spin systems
-\cite{wolff_comparison_1989, du_dynamic_2006, liu_dynamic_2014,
-wang_cluster_1990}, including the Ising, $\mathrm O(n)$
-\cite{wolff_collective_1989}, and Potts \cite{swendsen_nonuniversal_1987,
-baillie_comparison_1991} models. These algorithms rely on the natural symmetry
-of the systems in question under symmetry operations on the spins. Some
-success has been made in extending these algorithms to systems in certain
+timescales.
+
+Celebrated cluster algorithms largely addressed this in the absence of
+symmetry-breaking fields by using nonlocal updates \cite{janke_nonlocal_1998}
+whose clusters undergo a percolation transition at the critical point of the
+system \cite{coniglio_clusters_1980}.  These result in relatively small
+dynamic exponents for many spin systems \cite{wolff_comparison_1989,
+du_dynamic_2006, liu_dynamic_2014, wang_cluster_1990}, including the Ising,
+$\mathrm O(n)$ \cite{wolff_collective_1989}, and Potts
+\cite{swendsen_nonuniversal_1987, baillie_comparison_1991} models. These
+algorithms rely on the natural symmetry of the systems in question under
+symmetry operations on the spins.
+
+Some success has been made in extending these algorithms to systems in certain
 external fields by adding a `ghost site' \cite{coniglio_exact_1989} that
 returns global rotation invariance to spin Hamiltonians at the cost of an
 extra degree of freedom, allowing the method to be used in a subcategory of
@@ -95,16 +98,18 @@ using replica methods \cite{redner_graphical_1998, chayes_graphical_1998,
 machta_replica-exchange_2000}.  Monte Carlo techniques that involve cluster
 updates at fixed magnetization have been used to examine quantities at fixed
 field by integrating the associated thermodynamic functions
-\cite{martin-mayor_cluster_2009, martin-mayor_tethered_2011}.  We show that
-the scaling of correlation time near the critical point of several models
-suggests that the `ghost' approach is a natural one, e.g., that it extends the
-celebrated scaling of dynamics in these algorithms at zero field to various
-non-symmetric perturbations. We also show, by a redefinition of the spin--spin
-coupling in a generic class of spin systems, \emph{arbitrary} external fields
-can be treated using cluster methods. Rather than the introduction of a `ghost
-spin,\!' our representation relies on introducing a `ghost transformation.\!'
-
-\section{Introduction}
+\cite{martin-mayor_cluster_2009, martin-mayor_tethered_2011}.
+
+We show that the scaling of correlation time near the critical point of
+several models suggests that the `ghost' approach is a natural one, e.g., that
+it extends the celebrated scaling of dynamics in these algorithms at zero
+field to various non-symmetric perturbations. We also show, by a redefinition
+of the spin--spin coupling in a generic class of spin systems,
+\emph{arbitrary} external fields can be treated using cluster methods. Rather
+than the introduction of a `ghost spin,\!' our representation relies on
+introducing a `ghost transformation.\!'
+
+\section{Generalized 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 $V=\{1,\ldots,N\}$
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