AENC/resampling_chain Changeset - d4a402112276 · Centrum Wiskunde & Informatica (CWI)

Changeset - d4a402112276

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Tom Bannink - 8 years ago 2017-07-11 14:50:52
tom.bannink@cwi.nl

Fix typo

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 \begin{align*}
 	R^{(n)}(p) &= \frac{1}{n} \sum_{C\subseteq[n]} \sum_{f\in\{0,1'\}^{|C|}} \rho_{C(f)} R_{C(f)} ,
 \end{align*}
 where $C(f)\in\{0,1,1'\}^n$ denotes a configuration with slots on the sites $C$ filled with (anti)particles described by $f$. The non-slot positions are filled with $1$s.
 \begin{definition}[Diameter and gaps] \label{def:diameter} \label{def:gaps}
     For a subset $C\subseteq[n]$, we define the \emph{diameter} $\diam{C}$ to be the minimum size of an interval $I$ containing $C$. Here we consider both $C$ and the interval modulo $n$. In other words $\diam{C} = \min\{ j \vert \exists i : C\subseteq [i,i+j-1] \}$. We define the \emph{gaps} of $C$, as $I\setminus C$ and denote this by $\gaps{C}$. Note that $\diam{C} = |C| + |\gaps{C}|$.  Define $\maxgap{C}$ as the size of the largest connected component of $\gaps{C}$. Figure \ref{fig:diametergap} illustrates these concepts with a picture.
+    For a subset $C\subseteq[n]$, we define the \emph{diameter} $\diam{C}$ to be the minimum size of an integer interval $I$ containing $C$. Here we consider both $C$ and the interval modulo $n$. In other words $\diam{C} = \min\{ j \vert \exists i : C\subseteq [i,i+j-1] \}$. We define the \emph{gaps} of $C$, as $I\setminus C$ and denote this by $\gaps{C}$. Note that $\diam{C} = |C| + |\gaps{C}|$.  Define $\maxgap{C}$ as the size of the largest connected component of $\gaps{C}$. Figure \ref{fig:diametergap} illustrates these concepts with a picture.
 \end{definition}
 \begin{figure}
 	\begin{center}
     	\includegraphics{diagram_gap.pdf}
     \end{center}
     \caption{\label{fig:diametergap} Illustration of Definition \ref{def:diameter}. A set $C=\{1,2,4,7,9\}\subseteq[n]$ consisting of 5 positions is shown by the red dots. The smallest interval containing $C$ is $[1,9]$, so the diameter is $\diam{C}=9$. The blue squares denote the set $\gaps{C} = \{3,5,6,8\}$. The dotted line at the top depicts the rest of the circle which may be much larger. The largest gap of $C$ is $\maxgap{C}=2$ which is the largest connected component of $\gaps{C}$.}
     \hline
     \framebox{$1 0 1$} &\to \framebox{$0 0 0$} & p^3
 \end{align*}
 With this we can write a recursive formula for the expected number of resamples from $101$:
 \begin{align*}
     R_{101} &= (1-3p+3p^2 - p^3)(1) + (3p -6p^2 +3p^3) (1+R_{101}) \\
             &\quad + (p^2 - p^3) (1+R_{10101}) + (2p^2-2p^3) (1+R_{1001}) \\
+            &\quad + (p^2 - p^3) (1+R_{10101}) + (2p^2-2p^3) (1+R_{1001}) + p^3(1+R_{10001}) \\
 			&= 1 + 3 p + 7 p^2 + 14.6667 p^3 + 29 p^4 + 55.2222 p^5 + 102.444 p^6 + 186.36 p^7 \\
             &\quad + 333.906 p^8 + 590.997 p^9 + 1035.58 p^{10} + 1799.39 p^{11} + 3104.2 p^{12} \\
             &\quad+ 5322.18 p^{13} + 9075.83 p^{14} + 15403.6 p^{15} + 26033.4 p^{16} + 43833.5 p^{17} \\
             &\quad+ 73555.2 p^{18} + 123053 p^{19} + 205290 p^{20} + 341620 p^{21} + 567161 p^{22} \\
             &\quad+ 939693 p^{23} + 1.5537\cdot10^{6} p^{24} + 2.56158\cdot10^{6} p^{25} + \mathcal{O}(p^{26})
 \end{align*}

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