summaryrefslogtreecommitdiff
path: root/monte-carlo.tex
diff options
context:
space:
mode:
Diffstat (limited to 'monte-carlo.tex')
-rw-r--r--monte-carlo.tex28
1 files changed, 13 insertions, 15 deletions
diff --git a/monte-carlo.tex b/monte-carlo.tex
index 3faa4a5..bf76591 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -122,7 +122,7 @@ of $R$, e.g., for any $A\subseteq X$ and $r\in R$, $\mu(r\cdot A)=\mu(A)$.
This trait is shared by the counting measure on any discrete set, or by any
group acting by isometries on a Riemannian manifold, such as $\mathrm O(n)$ on
$S^{n-1}$ in the $\mathrm O(n)$ model \cite{caracciolo_wolff-type_1993}.
-Finally, the subset of elements in $R$ of order two must act transitively on
+Finally, a subset $R_2$ of elements in $R$ of order two must act transitively on
$X$. This property, while apparently obscure, is shared by any symmetric space
\cite{loos_symmetric_1969} or by any transitive, finitely generated isometry
group. In fact, all the examples listed here have spins spaces with natural
@@ -200,12 +200,10 @@ in the following way.
\begin{enumerate}
\item Pick a random site $m_0$ and add it to the stack.
- \item Select a rotation $r\in R$ of order two. For detailed balance it is
- sufficient to ensure that probability distribution $f_{m_0}(r\mid \set s)$ the
- rotation is sampled from depend only on $Z(s,r\cdot s)$ for every site,
- since these numbers are invariant under a cluster flip, e.g., $Z(s',r\cdot
- s')=Z(r\cdot s,r\cdot(r\cdot s))=Z(r\cdot s,(rr)\cdot s)=Z(r\cdot
- s,s)=Z(s,r\cdot s)$.
+ \item Select a rotation $r\in R_2$ distributed by $f(r\mid m_0,\set s)$.
+ Typically $f$ is uniform on $R_2$, but it is sufficient for preserving
+ detailed balance that $f$ be any function of the seed site $m_0$ and
+ $Z(s,r\cdot s)$ for all $s\in\set s$, which is what we will assume here.
\item While the stack isn't empty,
\begin{enumerate}
\item pop site $m$ from the stack.
@@ -224,14 +222,14 @@ in the following way.
When the stack is exhausted, a cluster of connected spins will have been
rotated by the action of $r$. In order for this algorithm to be useful, it
must satisfy ergodicity and detailed balance. 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
-probability $P(\set s\to\set{s'})$ that the configuration $\set s$ is brought
-to $\set s'$ by the flipping of a cluster formed by accepting rotations of
-spins via bonds $C\subseteq E$ and rejecting rotations via bonds $\partial
-C\subset E$ is related to the probability of the reverse process $P(\set{s'}\to\set s)$ by
+have ensured that the $R_2$ acts transitively on $X$, e.g., for any $s,t\in X$
+there exists $r\in R_2$ 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 probability $P(\set s\to\set{s'})$ that the
+configuration $\set s$ is brought to $\set s'$ by the flipping of a cluster
+formed by accepting rotations of spins via bonds $C\subseteq E$ and rejecting
+rotations via bonds $\partial C\subset E$ is related to the probability of the
+reverse process $P(\set{s'}\to\set s)$ by
\begin{widetext}
\[
\begin{aligned}