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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} |