From eb526148d370b2fbe96d3b049840c09a2ce44bf2 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 9 Nov 2017 12:30:40 -0500
Subject: some changes, also added the pdf

---
 monte-carlo.tex | 48 ++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 46 insertions(+), 2 deletions(-)

(limited to 'monte-carlo.tex')

diff --git a/monte-carlo.tex b/monte-carlo.tex
index 283b609..7133461 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -222,7 +222,7 @@ 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.
 A random site $i$ is selected from the graph and its spin is flipped. Each of the
-site's neighbors $j$ is also flipped with probability $1-e^{-2\beta|J_{ij}|}$
+site's neighbors $j$ is also flipped with probability $1-e^{-2\beta J_{ij}}$
 if doing so would lower the energy of the bond $i$--$j$. The process is
 repeated with every neighbor that was flipped. While this algorithm
 successfully addresses the problems of critical slowing down at zero field, it
@@ -233,7 +233,7 @@ then accepted or rejected Metropolis-style based on the change in energy
 same coupling to the external field, a strictly more efficient
 but exactly equivalent version of this hybrid is made by distributing the
 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
+added. In this version, the cluster is abandoned with probability $1-e^{-2\beta
 H}$ every time a spin is added to it.
 
 $z=0.29(1)$ \cite{wolff1989comparison,liu2014dynamic} $z=0.35(1)$ for
@@ -443,6 +443,10 @@ bottom plot is for simulations at }
   \input{fig_correlation_collapse-hL}
 \end{figure}
 
+Our algorithm for the Ising model in a field can be
+generalized to run on the $q$-spin Potts or $O(n)$ models in exactly the same
+way as the conventional Wolff algorithm.
+
 \section{Magnetization Estimator}
 
 At any size, the ensemble average magnetization $\avg M$ is identically zero
@@ -464,6 +468,46 @@ is no longer true. At zero field, for instance, it is common practice to use
 $\avg{|M|}$ to estimate the expected value for the magnetization instead of
 $\avg M$. But what to do at finite field? Is this approach justified?
 
+The provenance of the restricted configuration space in the thermodynamic
+limit is the fact that, at criticality or for nonzero $H$ under the critical
+temperature, an infinite cluster forms and can never be flipped by the
+dynamics. In a finite system there is no infinite cluster, and even the
+largest cluster has a nonzero probability of being flipped. However, we still
+want to be able to use finite systems to estimate the quantities of infinite
+ones. We approach this by thinking of a finite system as a piece of an
+infinite one. Clearly, time-averages of quantities in the chunk are equal to
+time-averages of the whole system, simply with much slower convergence.
+However, simulation of finite systems present a problem: in each sample, we
+are given a snapshot of a chunk of the system but do not know which direction
+the infinite cluster in the surrounding infinite system is pointing.
+Therefore, we must make a guess such that, in the limit of larger and larger
+system size, the probability that our guess is wrong and that the infinite
+cluster is pointed in a direction opposite the one we expected approaches
+zero. To estimate the equilibrium values of quantities in the infinite system,
+we only admit values resulting from states whose corresponding infinite
+cluster is expected to point in the direction of our external field. If the
+external field is zero, we choose one direction at random, say positive.
+
+Snapshots of the chunk what we suspect are likely to be a piece of an infinite
+cluster facing opposite the external field are discarded for the purpose of
+computing equilibrium values of the system, but still represent something
+physical. When the model's state contains an infinite cluster oriented against
+the external field, it is known as \emph{metastable}. Therefore in a finite
+simulation one can make convergent estimates of quantities in both the
+equilibrium and metastable states of the infinite system by dividing sampled
+states into two sets: those that are likely snapshots of a finite block in an
+infinite system whose infinite cluster is oriented with the field, and those
+whose infinite cluster is oriented against the field. All we must do now is
+specify how to distinguish between likely oriented with and likely oriented
+against. Others have used the direction of the largest cluster in the finite
+system to make this distinction. Far from the critical point in the
+low-temperature phase this does a clear and consistent job. However, we would
+like to study systems very near the critical point. We will take the likely direction of the
+infinite cluster to be given by \emph{the direction in which the system is
+magnetized}. Clearly, as our finite chunk grows larger and larger, the
+probability that its magnetization is different from the direction of the
+infinite cluster goes to zero. In the thermodynamic limite
+
 Since, in the thermodynamic limit expected values are given by an average
 over a restricted configuration space, we can estimate those expected values
 at finite size by making the same restriction. Defining the reduced
-- 
cgit v1.2.3-54-g00ecf

