From e7220fadb6b3775e55afd9a07831e90136cfaf24 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 7 Nov 2017 10:43:35 -0500
Subject: many changes, including added references

---
 monte-carlo.tex | 48 ++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 40 insertions(+), 8 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index fe2a315..2490b31 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -82,7 +82,7 @@
 
 \begin{document}
 
-\title{An efficient cluster algorithm for the Ising model in an external field}
+\title{Efficiently sampling Ising states in an external field}
 \author{Jaron Kent-Dobias}
 \author{James P.~Sethna}
 \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA}
@@ -90,15 +90,43 @@
 \date\today
 
 \begin{abstract}
-  An abstract.
+  We introduce an extension of the Wolff algorithm that preforms efficiently
+  in an external magnetic field. Near the Ising critical point, the
+  correlation time of our algorithm has a conventional scaling form that
+  reduces to that of the Wolff algorithm at zero field. As an application, we
+  directly measure scaling functions of observables in the metastable state of
+  the 2D Ising model.
 \end{abstract}
 
 \maketitle
 
-\section{Introduction}
-
 The Ising model is a simple model of a magnet comprised of locally interacting
-spins.
+spins. Like most large thermal systems, computation of its properties cannot
+be carried out explicitly and is preformed using Monte Carlo methods. Near its
+continuous phase transition, divergent correlation length leads to divergent
+correlation time in any locally-updating algorithm, hampering computation.
+At zero external field, this was largely alleviated by cluster algorithms,
+like the Wolff algorithm, whose dynamics are nonlocal and each step flips
+groups of spins whose size diverges with the correlation length. However, the
+Wolff algorithm only works at zero field. We describe an extension of this
+algorithm that works in arbitrary external field while preserving the Wolff
+algorithm's small dynamic exponent.
+
+The Wolff algorithm works by first choosing a random spin and adding it to an
+empty cluster. Every neighbor of that spin pointed in the same direction as
+the spin is added to the cluster with probability $1-e^{-2\beta J}$, where
+$\beta=1/T$ and $J$ is the coupling between sites. This process is iterated
+again for neighbors of every spin added to the cluster. When all sites
+surrounding the cluster have been exhausted, the cluster is flipped. Our
+algorithm is a simple extension of this. An extra spin is introduced (often
+referred to as a ``ghost spin'') that couples to all others with coupling $H$.
+The traditional Wolff algorithm is then preformed on this larger lattice exactly as described above,
+with the extra spin treated no differently from any others. Observables in the
+original system can be exactly estimated on the new one using a simple
+mapping. As an application, we use our algorithm to measure critical scaling functions
+of the 2D Ising model in its metastable phase.
+
+\section{Introduction}
 
 Consider an undirected graph $G=(V,E)$ describing a system of interacting spins. The set
 of vertices $V=\{1,\ldots,N\}$ enumerates the sites of the network, and the
@@ -147,7 +175,8 @@ $S^n$ according to the Boltzmann distribution $e^{-\beta\H(s)}$, so that
 averages of observables made using their samples asymptotically approach the
 true expected value.
 
-The Metropolis--Hastings algorithm is very popular for systems in statistical
+The Metropolis--Hastings algorithm
+\cite{metropolis1953equation,hastings1970monte} is very popular for systems in statistical
 physics. A starting state $s\in S^n$ is randomly perturbed to the state $s'$,
 usually by flipping one spin. The change in energy $\Delta\H=\H(s')-\H(s)$ due
 to the perturbation is then computed. If the change is negative the perturbed
@@ -177,7 +206,7 @@ take many perturbations to move between in configuration space.
   \end{algorithm}
 \end{figure}
 
-The Wolff algorithm solves many of these problems, but only at zero external
+The Wolff algorithm \cite{wolff1989collective} solves many of these problems, but only at zero external
 field, $H=0$. This algorithm solves the problem of critical slowing-down by
 flipping carefully-constructed clusters of spins at once in a way that samples
 high-correlated states quickly while also always accepting prospective states.
@@ -196,6 +225,9 @@ cluster acceptance or rejection across the spins in the cluster as they are
 added. In this version, the cluster is abandoned with probability $1-e^{-\beta
 H}$ every time a spin is added to it.
 
+$z=0.29(1)$ \cite{wolff1989comparison,liu2014dynamic} $z=0.35(1)$ for
+Swendsen--Wang \cite{swendsen1987nonuniversal}
+
 \begin{figure}
   \begin{algorithm}[H]
     \begin{algorithmic}
@@ -491,7 +523,7 @@ a given average if the associated state is in the reduced space of interest.
   Thanks!
 \end{acknowledgments}
 
-%\bibliography{monte-carlo}
+\bibliography{monte-carlo}
 
 \end{document}
 
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