From f99db951a095b0f49cba7fe9f74000829900487d 2017-09-07 17:27:27
From: Tom Bannink <tom.bannink@cwi.nl>
Date: 2017-09-07 17:27:27
Subject: [PATCH] Change figure position to wrapfigure

---

diff --git a/main.tex b/main.tex
index be357c0b9e0687d57fee5f2a020f791f45525c79..3fd54e4e66423452243104cdd0fdce3c38ac0c3f 100644
--- a/main.tex
+++ b/main.tex
@@ -9,6 +9,7 @@
 \usepackage{diagbox}
 \usepackage[table]{xcolor}% http://ctan.org/pkg/xcolor
 \usepackage{graphicx}
+\usepackage{wrapfig}
 \usepackage{caption}
 \captionsetup{compatibility=false}
 \graphicspath{{./}}
@@ -453,13 +454,18 @@ The process on the finite chain has the following modification at the boundary:
 \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 $v,w$.}
-\end{figure}
-\begin{lemma}[Conditional independence] \label{lemma:eventindependence} \label{claim:eventindependence}
+%\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 $v,w$.}
+%\end{figure}
+\begin{wrapfigure}{r}{0.25\textwidth}
+    \centering
+    \includegraphics{diagram_groups.pdf}
+    \caption{\label{fig:separatedgroups} Lemma \ref{lemma:eventindependence}.}
+\end{wrapfigure}
+The following lemma considers two vertices $v,w$ that are never ``crossed'' so that two halves of the cycle become independent.\begin{lemma}[Conditional independence] \label{lemma:eventindependence} \label{claim:eventindependence}
     Let $b=b_1\land b_2\in\{0,1\}^n$ be a state with two groups of zeroes that are separated by at least one site inbetween, as in Figure \ref{fig:separatedgroups}. Let $v$, $w$ be any indices inbetween the groups, such that $b_1$ lies on one side of them and $b_2$ on the other, as shown in the figure. Furthermore, let $A_1$ be any event that depends only on the sites ``on the $b_1$ side of $v,w$'', and similar for $A_2$ (for example $\mathrm{Z}^{(i)}$ for an $i$ on the correct side). Then we have
     \begin{align*}
         \P^{(n)}_b(\mathrm{NZ}^{(v,w)}, A_1, A_2)
@@ -480,7 +486,6 @@ The process on the finite chain has the following modification at the boundary:
     \end{align*}
     %up to any order in $p$.
 \end{lemma}
-The lemma says that conditioned on $v$ and $w$ not being crossed, the two halves of the cycle are independent. 
 
 \begin{proof}
     From any path $\xi\in\start{b} \cap \mathrm{NZ}^{(v,w)}$ we can construct paths $\xi_1\in\start{b_1}\cap \mathrm{NZ}^{(v,w)}$ and $\xi_2\in\start{b_2}\cap\mathrm{NZ}^{(v,w)}$ as follows. Let us write the path $\xi$ as