From 67a5850a6c6e27d7722483de55912df293b63a1a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 2 Oct 2018 20:08:00 -0400
Subject: introduced R_2, a subset of R of order two

---
 monte-carlo.tex | 28 +++++++++++++---------------
 1 file changed, 13 insertions(+), 15 deletions(-)

(limited to 'monte-carlo.tex')

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}
-- 
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