Changeset - e62754c9120c
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Tom Bannink - 8 years ago 2017-09-14 15:45:53
tom.bannink@cwi.nl
Add definition of local event
1 file changed with 11 insertions and 5 deletions:
main.tex
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@@ -51,7 +51,7 @@
 
\def\im{\mathrm{im}}
 
\newcommand{\bOt}[1]{\widetilde{\mathcal O}\left(#1\right)}
 
\newcommand{\bigO}[1]{\mathcal O\left(#1\right)}
 
\newcommand{\Res}[1]{\#\mathrm{Res}\left(#1\right)}
 
\newcommand{\Res}[1]{\#\textsc{Res}\left(#1\right)}
 

	
 
\newcommand{\QMAo}{\textsf{QMA$_1$}}
 
\newcommand{\BQP}{\textsf{BQP}}
 
@@ -257,10 +257,16 @@ We consider the following generalization of the Markov Chain.
 
Let $G=(V,E)$ be an undirected graph with vertex set $V$ and edge set $E$. We define a Markov Chain $\mathcal{M}_G$ as the following process: initialize every vertex of $G$ independently to 0 with probability $p$ and 1 with probability $1-p$. Then at each step, select a uniformly random vertex that has value $0$ and resample it and its neighbourhood, all of them independently with the same probability $p$. The Markov Chain terminates when all vertices have value $1$. We use $\P^{G}$ to denote probabilities associated to this Markov Chain and $\E^G$ to denote expectation values.
 

	
 
\begin{definition}[Events and notation] \label{def:events}
 
    Let $G=(V,E)$ be a graph. Let $S\subseteq V$ be any subset of vertices.
 
    Let $G=(V,E)$ be a graph. Let $S\subseteq V$ be any subset of vertices, and let $v\in V$ be any vertex.
 
    \begin{itemize}
 
        \item Define $\NZ{S}$ as the event that \emph{none} of the vertices in $S$ become zero at any point in time before the Markov Chain terminates.    	
 
        \item Define $\Z{S}$ as the complement of $\NZ{S}$, i.e. the event that \emph{there exists} a vertex in $S$ that becomes zero at some point in time before the Markov Chain terminates.
 
        \item Let $\Res{v}$ be the number of times that $v$ was picked as a center of resampling.
 
        \item We say an event $A$ is \emph{local} on the vertex set $S$ if it is in the sigma algebra generated by the events
 
            \begin{align*}
 
                \NZ{v} \; , \; \Z{v}\cap(\Res{v}=0) \; , \; (\Res{v} = k)
 
            \end{align*}
 
            for all $v\in S$ and $1\leq k \leq \infty$.
 
        \item Define for any event $A$:
 
            \begin{align*}
 
                \P^{G}_S(A) &= \P^{G}(A \mid \text{All vertices in $S$ get initialized to }1)
 
@@ -422,9 +428,9 @@ The intuition of the following lemma is that if two sites have distance $d$ in t
 
	Now we use the Splitting lemma~\ref{lemma:splitting} to show that for all $i\in[d]$
 
	\begin{align}
 
		\P^G(A_{i}^X)
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \P^{G\setminus B(X,i-1)}(X,i)(\NZ{\overline{\partial}(X,i)}) \tag{by Lemma~\ref{lemma:splitting}}\\
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \left(\P^{G\setminus Y\setminus B(X,i-1)}(X,i)(\NZ{\overline{\partial}(X,i)})+\bigO{p^{d+1-i}}\right) \tag{by induction}\\
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \P^{G\setminus Y\setminus B(X,i-1)}(X,i)(\NZ{\overline{\partial}(X,i)})+\bigO{p^{d+1}} \tag{by equation \eqref{eq:AXorder}}\\
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \P^{G\setminus B(X,i-1)}(\NZ{\overline{\partial}(X,i)}) \tag{by Lemma~\ref{lemma:splitting}}\\
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \left(\P^{G\setminus Y\setminus B(X,i-1)}(\NZ{\overline{\partial}(X,i)})+\bigO{p^{d+1-i}}\right) \tag{by induction}\\
 
		&=\P^{G\cap B(X,i)}_{\overline{\partial}(X,i)}(A_{i}^X)\cdot \P^{G\setminus Y\setminus B(X,i-1)}(\NZ{\overline{\partial}(X,i)})+\bigO{p^{d+1}} \tag{by equation \eqref{eq:AXorder}}\\
 
		&=\P^{G\setminus Y}(A_{i}^X)+\bigO{p^{d+1}} \tag{by Lemma~\ref{lemma:splitting}}\\
 
		&=\P^{G\setminus Y}(A_{i}^X)+\bigO{p^{d(Y,Y)}}. \label{eq:indStep}
 
	\end{align}
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