added info on averaging expectation values

1 files changed, 20 insertions, 12 deletions

diff --git a/monte-carlo.tex b/monte-carlo.tex
index c1c70ab..23ce92b 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -157,7 +157,7 @@ invariant under the action of any element of $R$ applied to the entire lattice,
 $s,t\in X$, $Z(r\cdot s,r\cdot t)=Z(s,t)$. 
 One may also allow $Z$ to also be a function of the edge---for modelling
 random-bond, long-range, or anisotropic interactions---or allow $B$ to be a
-function of site---for modelling random fields. All the formal results of this paper hold equally
+function of site---for applying arbitrary boundary conditions or modelling random fields. All the formal results of this paper hold equally
 well for these cases, but we will drop the additional index notation for clarity.
 
 \begin{table*}[htpb]
@@ -198,7 +198,18 @@ that state appearing, or
 \]
 where for $Y_1\times\cdots\times Y_N=Y\subseteq X^N$ the measure
 $\mu(Y)=\mu(Y_1)\cdots\mu(Y_N)$ is the simple extension of the
-measure on $X$ to a measure on $X^N$.
+measure on $X$ to a measure on $X^N$. These values are estimated by Monte
+Carlo techniques by constructing a finite sequence of states $\{\vec
+s_1,\ldots,\vec s_M\}$ such that
+\[
+  \avg A\simeq\frac1M\sum_{i=1}^MA(\vec s_i)
+\]
+Sufficient conditions for this average to converge to $\avg A$ as $M\to\infty$
+are that the process that selects $\vec s_{i+1}$ given the previous states be
+Markovian (only depends on $\vec s_i$), ergodic (any state can be accessed),
+and obey detailed balance (the ratio of probabilities that $\vec s'$ follows
+  $\vec s$ and vice versa is equal to the ratio of weights for $\vec s$ and
+$\vec s'$ in the ensemble).
 
 While any several related cluster algorithms can be described for this
 system, we will focus on the Wolff algorithm in particular
@@ -252,15 +263,11 @@ probability of the reverse process $P(\vec s'\to\vec s)$ by
 \end{aligned}
 \]
 \end{widetext}
-which satisfies detailed balance so long as $p_r(s,t)=p_{r^{-1}}(s,t)$.
-Therefore, for the algorithm to function one must choose rotations at random
-only from those group elements in $R$ that act by idempotents. Usually, as is
-the case in all the examples here, the action is natural enough that this is
-the same as only choosing group elements of order two to act on your spins.
-Ergodicity is achieved if the subset of symmetry rotations that obey this
-property allow any spin state to access any other under composition. This is
-true for all the
-models in Table \ref{table:models}.
+whence detailed balance is satisfied. Ergodicity is satisfied since we have
+ensured that the subset of elements in $R$ that are order two acts
+transitively on $K$, e.g., for any $s,t\in X$ there exists $r\in R$ such that
+$r\cdot s=t$. Since there is a nonzero probability that only one spin is
+rotated and that spin can be rotated into any state, ergodicity follows.
 
 The function of the algorithm described above depends on the fact that the
 coupling $Z$ depends only on the relative orientation of the spins---global
@@ -433,7 +440,7 @@ $s_i$, which suffice to provide ergodicity as any integer can be taken to any
 other in one step of this kind. The coupling is usually taken to be
 $Z(i,j)=(i-j)^2$, though it may also be any function of the absolute
 difference $|i-j|$.
-Because randomly choices of integer will almost always result in energy
+Because random choices of integer will almost always result in energy
 changes so big that the whole system is always flipped, it is better to select
 random reflections about integers close to the average state of the system.
 Continuous roughening models---where the spin states are described by real
@@ -475,6 +482,7 @@ We measured the autocorrelation time using methods here
   }
 \end{figure}
 
+
 \begin{acknowledgments}
 \end{acknowledgments}