# How to calculate the exact probability $p$ at which maxima occurs in the curves in an infinite system?

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

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.)
Is there any method to exactly calculate the $p$, for which the maxima of the curves (of black clusters, and white clusters) occur when $L\to\infty$ ?
• 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. – user119567 Mar 11 '18 at 21:08
• 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? – Carlo Beenakker Mar 11 '18 at 22:17
• 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. – Robert Israel Mar 12 '18 at 3:21