\newcommand{\pvp}{\vec{p}{\kern 0.45mm}'}

\DeclarePairedDelimiter\bra{\langle}{\rvert}

\DeclarePairedDelimiter\ket{\lvert}{\rangle}

\DeclarePairedDelimiterX\braket[2]{\langle}{\rangle}{#1 \delimsize\vert #2}

\newcommand{\underflow}[2]{\underset{\kern-60mm \overbrace{#1} \kern-60mm}{#2}}

\def\Ind(#1){{{\tt Ind}({#1})}}

\def\Id{\mathrm{Id}}

\def\Pr{\mathrm{Pr}}

\def\Tr{\mathrm{Tr}}

\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{\QMAo}{\textsf{QMA$_1$}}

\newcommand{\BQP}{\textsf{BQP}}

\newcommand{\NP}{\textsf{NP}}

\newcommand{\SharpP}{\textsf{\# P}}

\newcommand{\diam}[1]{\mathcal{D}\left(#1\right)}

\newcommand{\paths}[1]{\mathcal{P}\left(#1\to\mathbf{1}\right)}

\newcommand{\start}[1]{\textsc{start}\left(#1\right)}

\newcommand{\maxgap}[1]{\mathrm{maxgap}\left(#1\right)}

\newcommand{\gaps}[1]{#1_{\mathrm{gaps}}}

\renewcommand{\P}{\mathbb{P}}

\newcommand{\E}{\mathbb{E}}

\newcommand{\NZ}[1]{\mathrm{NZ}^{(#1)}}

\newcommand{\Z}[1]{\mathrm{Z}^{(#1)}}

%\newcommand{\dist}[1]{d_{\!\!\not\,#1}}

\newcommand{\dist}[1]{d_{\neg #1}}

\newcommand{\todo}[1]{{\color{red}\textbf{TODO:} #1}}

\long\def\ignore#1{}

\newtheorem{theorem}{Theorem}

\newtheorem{corollary}[theorem]{Corollary}%[theorem]

\newtheorem{lemma}[theorem]{Lemma}

\newtheorem{prop}[theorem]{Proposition}

\newtheorem{definition}[theorem]{Definition}

\newtheorem{claim}[theorem]{Claim}

\newtheorem{remark}[theorem]{Remark}

\newenvironment{proof}

{\noindent {\bf Proof. }}

{{\hfill $\Box$}\\	\smallskip}

		&= \sum_{k=1}^{\infty}\sum_{P\text{ patch}:1\in P}\P^{[-N,N]}(\Res{1}\geq k\,\&\, P\in\mathcal{P}) +\bigO{p^{n}} \tag{by Lemma~\ref{lemma:eventindependenceNew}}\\

		&= \E^{[-N,N]}(\Res{1})+\bigO{p^{n}}.

		\end{align*}	

\end{comment}			

~

Questions:

\begin{itemize}

	\item Can we generalise the proof to other translationally invariant spaces, like the torus?

	\item In view of this proof, can we better characterise $a_k^{(k+1)}$?

	\item Why did Mario's and Tom's simulation show that for fixed $C$ the contribution coefficients have constant sign? Is it relevant for proving \ref{it:pos}-\ref{it:geq}?

\end{itemize} 

	%I think the same arguments would translate to the torus and other translationally invariant spaces, so we could go higher dimensional as Mario suggested. Then I think one would need to replace $|S_{><}|$ by the minimal number $k$ such that there is a $C$ set for which $S\cup C$ is connected. I am not entirely sure how to generalise Lemma~\ref{lemma:probIndep} though, which has key importance in the present proof.

\newpage

\section{General graphs proof}

We consider the following generalization of the Markov Chain.

Let $G=(V,E)$ be a 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.

\begin{definition}[Events] \label{def:events}

    \begin{align*}

    \end{align*}

\end{definition}

$\NZ{S}$

$\Z{S}$

patch

$B(S;d)$

We consider $R^{(n)}(p)$ as a power series in $p$ and our main aim in this section is to show that $R^{(n)}(p)$ and $R^{(n+k)}(p)$ are the same up to order $n-1$.

%Note that we have $\P^{(n)}(\start{b}) = (1-p)^{|b|}p^{n-|b|}$ by definition of our Markov Chain.

\begin{definition}[Vertex visiting event] \label{def:visitingResamplings}

    Denote by $\mathrm{Z}^{(v)}$ the event that site $v$ becomes zero at any point in time before the Markov Chain terminates. Denote the complement by $\mathrm{NZ}^{(v)}$, i.e. the event that site $v$ does \emph{not} become zero before it terminates. Furthermore define $\mathrm{NZ}^{(v,w)} := \mathrm{NZ}^{(v)} \cap \mathrm{NZ}^{(w)}$, i.e. the event that \emph{both} $v$ and $w$ do not become zero before termination.

\end{definition}

%\begin{figure}

%	\begin{center}

%    	\includegraphics{diagram_groups.pdf}

%    \end{center}

%    \caption{\label{fig:separatedgroups} Illustration of setup of Lemma \ref{lemma:eventindependence}. Here $b_1,b_2\in\{0,1\}^n$ are bitstrings such that all zeroes of $b_1$ and all zeroes of $b_2$ are separated by two indices $v,w$.}