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Draft: Resolve "autocorrelation time" #566

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26 changes: 13 additions & 13 deletions Documentation/Doc.idx
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\indexentry{\texttt{Checkerboard}}{25}
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\indexentry{\path{Bilayer_square}}{78}
\indexentry{\path{N_leg_ladder}}{78}
\indexentry{\path{Honeycomb}}{78}
\indexentry{\texttt{Bilayer\_honeycomb}}{78}
\indexentry{\texttt{Triangular}}{79}
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2 changes: 1 addition & 1 deletion Documentation/sampling.tex
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Expand Up @@ -74,7 +74,7 @@ \subsection{An explicit example of error estimation}\label{sec:autocorr}
In order to determine the autocorrelation time, we calculate the correlation function
\begin{equation}
\label{eqn:autocorrel}
S_{\hat{O}}(t_{\textrm{Auto}})=\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\frac{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
S_{\hat{O}}(t_{\textrm{Auto}})=\frac{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i+t_{\textrm{Auto}}}-\left\langle \hat{O}\right\rangle \right)}{\sum_{i=1}^{N_{\textrm{Bin}}-t_{\textrm{Auto}}}\left(O_i-\left\langle \hat{O}\right\rangle \right)\left(O_{i}-\left\langle \hat{O}\right\rangle \right)}\, ,
\end{equation}
where $O_i$ refers to the Monte Carlo estimate of the observable $\hat{O}$ in the $i^{\text{th}}$ bin. This function typically shows an exponential decay and the decay rate defines the autocorrelation time.
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