AENC/resampling_chain Changeset - 77af5953ea57 · Centrum Wiskunde & Informatica (CWI)

Changeset - 77af5953ea57

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Tom Bannink - 8 years ago 2017-09-08 14:25:49
tom.bannink@cwi.nl

Merge branch 'master' of https://git.overleaf.com/9473097rgxjnfdkjdhb

1 file changed with 6 insertions and 3 deletions:

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 \end{proof}
 	\begin{definition}[Connected patches]
-		Let $P$ be an interval $[a,b]$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $P\in \mathcal{P}$ denote the event that one of the patches is equal to $P$.
+		Let $P\subseteq V$ be a connected component of $G$. We say that $P$ is a patch of a particular run of the process if $P$ is a maximal connected component of the vertices that have ever become $0$ before termination. We denote the set of patches of a run by $\mathcal{P}$. For a patch $P$ let $P\in \mathcal{P}$ denote the event that one of the patches is equal to $P$.
 		In other words
 		\begin{align*}
 		P\in\mathcal{P} := \NZ{a-1} \cap \Z{a} \cap \Z{a+1} \cap \cdots \cap \Z{b-1} \cap \Z{b} \cap \NZ{b+1} .
 		P\in\mathcal{P} := \NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}
 		For $\mathcaP{I}'\subseteq 2^{2^V}$ a set of patches we denote by $\mathcal{P}'\in \mathcal{P}$ the event that $\mathcal{P}'$ is a subset of the patches, i.e.,
 		\begin{align*}
 			\mathcal{P}'\in \mathcal{P} := \bigcup_{P\in \mathcal{P}'}\NZ{\overline{\partial}P} \cap \Z{P}.
 		\end{align*}
 		(In the extreme case when $P$ covers the whole cycle $[n]$, then instead $P\in\mathcal{P}:= \bigcap_{v\in[n]}\Z{v}$.)
 	\end{definition}
 	We are often going to use the observation that we can partition the event $\Z{v}$ using patches:

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