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diff --git a/monte-carlo.tex b/monte-carlo.tex index 265e376..db606a1 100644 --- a/monte-carlo.tex +++ b/monte-carlo.tex @@ -500,7 +500,7 @@ When $x<1$ transformations that do not change the energy of a bond can still activate it in the cluster, which allows nontrival clusters to be seeded when the height of the starting site is also the plane of reflection. This modification is likely useful in general for systems with large yet discrete -configuration spaces $X$. +state spaces. \section{Performance} \label{sec:performance} @@ -509,13 +509,13 @@ No algorithm is worthwhile if it doesn't run efficiently. This algorithm, being an extension of the Wolff algorithm into a new domain, should be considered successful if it likewise extends the efficiency of the Wolff algorithm into that domain. Some systems are not efficient under Wolff, and we -don't expect this extension to help them. For instance, Ising models with -random fields or bonds technically can be treated with Wolff +don't expect them to fare better when extended in a field. For instance, Ising +models with random fields or bonds technically can be treated with Wolff \cite{dotsenko_cluster_1991}, but it is not efficient because the clusters -formed do scale naturally with the correlation length -\cite{rieger_monte_1995,redner_graphical_1998}. Other approaches, like replica methods, should -be relied on instead -\cite{redner_graphical_1998,chayes_graphical_1998,machta_replica-exchange_2000}. +formed do scale naturally with the correlation length \cite{rieger_monte_1995, +redner_graphical_1998}. Other approaches, like replica methods, should be +relied on instead \cite{redner_graphical_1998, chayes_graphical_1998, +machta_replica-exchange_2000}. At a critical point, correlation time $\tau$ scales with system size $L=N^{-D}$ as $\tau\sim L^z$. Cluster algorithms are celebrated for their @@ -528,18 +528,25 @@ as a function of temperature $t$ and field $h$ of the form If a given dynamics for a system at zero field results in scaling like $L^z$, one should expect its natural extension in the presence of a field to scale roughly like $h^{-z\nu/\beta\delta}$ and collapse appropriately as a function -of $hL^{\beta\delta/\nu}$. We measured the autocorrelation time for the $D=2$ -square-lattice model at a variety of system sizes, temperatures, and fields -$B(s)=hs/\beta$ using standard methods \cite{ossola_dynamic_2004}. The -resulting scaling behavior, plotted in +of $hL^{\beta\delta/\nu}$. + +We measured the autocorrelation time of the energy $\H$ for a variety of +models at critical temperature with many system sizes and canonical fields +(see Table~\ref{table:models} with $h=\beta H$) using standard methods +\cite{ossola_dynamic_2004}. The resulting scaling behavior, plotted in Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with an -extension to finite field of the behavior at zero field. +extension to finite field of the behavior at zero field, with an eventual +finite-size crossover to constant autocorrelation time. \begin{figure*} \include{fig_correlation-times} \caption{ - Correlation times $\tau$ scaled by the average cluster size as a function - of external field for various models at various system sizes. + Scaling collapse of autocorrelation times $\tau$ for the energy $\H$ + scaled by the average cluster size as a function of external field for + various models of Table~\ref{table:models}. Critical exponents are + model-dependent. Colored lines and points depict values as measured by the + extended algorithm. Solid black lines show a plot proportional to + $h^{-z\nu/\beta\delta}$ for each model. } \label{fig:correlation_time-collapse} \end{figure*} |