From a14984cd2c086d013eaf785a7faa237fe331abde 2017-09-08 17:10:16
From: Andras Gilyen <gilyen@clayoquot.swat.cwi.nl>
Date: 2017-09-08 17:10:16
Subject: [PATCH] Incr.

---

diff --git a/main.tex b/main.tex
index ff16acc0fb35a77cad5a0a42b7d3e5b92c9f6723..5c66397e30fc5e62d186c5943c89b5bddfaac80c 100644
--- a/main.tex
+++ b/main.tex
@@ -61,7 +61,6 @@
 \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{\initone}[1]{\textsc{InitOne}\left(#1\right)}
 \newcommand{\patch}[1]{\textsc{Patch}\left(#1\right)}
 \newcommand{\patches}[1]{\textsc{Patches}\left(#1\right)}
 \newcommand{\maxgap}[1]{\mathrm{maxgap}\left(#1\right)}
@@ -593,41 +592,22 @@ Questions:
 
 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.
+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 and $\E^G$ to denote expectation values.
 
 \begin{definition}[Events] \label{def:events}
     Let $S\subseteq V$ be any subset of vertices.
-    Define $\Z{S}$ as the event that \emph{all} vertices in $S$ become zero at any point in time before the Markov Chain terminates.
-    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.
-    Define $\initone{S}$ as the event that all vertices in $S$ \emph{initially} get assigned the value 1, and define for any event $A$:
-    \begin{align*}
-        \P^{G}_S(A) &= \P^{G}(A \;\mid\; \initone{S})
-    \end{align*}
+    \begin{itemize}
+        \item Define $\Z{S}$ as the event that \emph{all} vertices in $S$ become zero at any point in time before the Markov Chain terminates.
+        \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 for any event $A$:
+            \begin{align*}
+                \P^{G}_S(A) &= \P^{G}(A \mid \text{All vertices in $S$ get initialized to }1)
+            \end{align*}
+        \item Boundary $\partial$ \todo{}
+        \item $d$-Neighbourhood $B(S;d)$ \todo{}
+    \end{itemize}
 \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:visitingResamplingsGen}
-    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:eventindependenceGen}. 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 $v,w$.}
-%\end{figure}
 \begin{wrapfigure}[7]{r}{0.25\textwidth} % The first [] argument is number of lines that are narrowed
     \centering
     \includegraphics{diagram_groups.pdf}