I'm writing with respect to this paper: Khatun, Dutta, and Tarafdar - "Islands in Sea" and "Lakes in Mainland" phases and related transitions simulated on a square lattice

Here's a link to the PDF version.

The following graph shows the number of clusters (black and white respectively) per unit area for any $p\in [0,1]$, where $p$ is the probability of a square being black, for different system sizes ($L=8/16/32/64/128/256$ units). (Please read the abstract I linked for a brief explanation of the type of system we're dealing with.)

enter image description here

Is there any method to exactly calculate the $p$, for which the maxima of the curves (of black clusters, and white clusters) occur when $L\to\infty$ ?

Yes, I know that I can do numerical testing and then use extrapolation, but I don't want to do that as extrapolation often predicts values with large amount of error.

  • $\begingroup$ If it's not possible to exactly obtain the limiting value, I'd be interested in knowing the mathematical reason as to why it will be not possible. Maybe I should mention that in the question? @CarloBeenakker $\endgroup$ – user119567 Mar 11 '18 at 21:04
  • $\begingroup$ I've computed the graphs for $L=1000,5000,10000$ as well. All of them seem to maintain the sigmoidal nature. So just from intuition (nothing else), I think the limiting graph would have a similar nature with a maxima somewhere before $p=0.5$ for the black cluster graph and somewhere after $p=0.5$ for the white cluster graph. $\endgroup$ – user119567 Mar 11 '18 at 21:08
  • $\begingroup$ May I may please know the reason for the downvote? $\endgroup$ – user119567 Mar 11 '18 at 21:18
  • 1
    $\begingroup$ I did not downvote, but I don't quite understand what type of answer you are hoping for: in the cited paper the functional form for the maximum is assumed to be $p_{\rm max}(L)=p_{\infty}+c/L$, and a fit gives the two parameters $p_{\infty}$ and $c$, resulting in a $L\rightarrow\infty$ limit of $p_\infty= 0.165$ for the black cluster and $p_\infty= 0.839$ for the white cluster; what else is there to say about this problem? $\endgroup$ – Carlo Beenakker Mar 11 '18 at 22:17
  • 1
    $\begingroup$ In general there are very few models of this sort where $L \to \infty$ limits can be computed exactly, unless they are either trivial (say $p=0$ or $p=\infty$) or determined by a symmetry. $\endgroup$ – Robert Israel Mar 12 '18 at 3:21

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.