How ``flat'' is ``flat''?

In this section, we report results of a small case study which investigates the required flatness criterion to achieve a converged density of states. We will consider the converged $\Omega$ computed using the 80% criterion as the ``exact'' result, and compare this to $\Omega$'s computed using the criterion that every bin in the energy histogram has at least $M$ hits. We considered $M \in \left\{2,20,200,2000,20000,200000\right\}$. In order to minimize the number of cycles, we will sample the histogram (that is, query its flatness) once per cycle. We also used the same termination criterion for the update of $f$. In the figure below, I show both the relative error between $\Omega$ computed with the $M$-criterion and $\Omega_{\rm exact}$, as well as the number of cycles required for the $M$-criterion computation.

Relative error ($\Box$) and number of cycles ($\bigcirc$) vs. minimum hit level $M$ for flatness in WL sampling of the $L$ = 12 ten-state Potts model.

Interestingly, we see that even out to $>10^6$ cycles, our result has only marginally converged to the exact result; an error of $\approx 0.1$ is larger than the iteration-to-iteration convergence tolerance of $\approx 10^{-4}$ observed in the original run. Nevertheless, this level of error is very small; the curves are indistinguishable on a plot. Another interesting fact is that the $M$ = 2 run is apparently more expensive than the $M$ = 20 run. This is likely due to fact that the initial configuration for each run segment (each value of $f$) is different; it is the final configuration from the previous segment. It just so happens that the 23rd iteration for $M$ = 2 begins in a configuration that initiates a walk that more slowly covers energy space than the configurations in the $M$ = 20 runs. Beyond $M$ = 20, the behavior is as expected: increasing $M$ makes the simulation more expensive.