I was looking at this paper “Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice on Percolation theory. The concept of phase transition used here seems to be a bit different compared to the more popular bond percolation or site percolation based models of phase transition. Now, normally for phase transitions critical exponents exist. I wonder if we consider the probability $p$ analogous to temperature, there is some physical quantity $f$, which can be expressed in terms of a power law close to $p_{c1}$ and $p_{c2}$.

In the paper they do show that $(d\chi/dp)^{-1}$ vs. $|p-p_{c1}|$ or $(d\chi/dp)^{-1}$ vs. $|p-p_{c2}|$, follows *sort of* a power law. But it doesn't seem to follow the power law exactly. And secondly $\chi$ isn't a very *physical* quantity so as to speak. It would be more interesting if something like correlation length followed a power law.

So, my question is basically:Has there been any research based on a similar model of phase transition, which shows existence of a critical exponent other than the one shown in this paper (i.e. $(d\chi/dp)^{-1}$ vs. $|p-p_{c1}|$ or $(d\chi/dp)^{-1}$ vs. $|p-p_{c2}|$) for the infinite system case (for a $N\times N$ system where $N\to\infty$)? Also, I'd be interested if someone on MathOverflow could suggest a $f$ which might possibly follow a power law during phase transition, or prove their existence.

Edit 1: As requested in the comments I'm adding the definitions of the variables used in the question.

$\chi(p)$ has been defined as $N_B(p)-N_W(p)$ where $N_B(p)$ and $N_W(p)$ are the number of black clusters and white clusters respectively at a probability $p$. Although it is called "Euler's number" I doubt it is a true invariant (which is a requirement in mathematical topology).

$p_{c1}$ is the probability $p$, when transition from $N_W=1$ to $N_W>1$ occurs (white background

*breaks*).$p_{c_2}$ is the probability $p$, when transition from $N_B>1$ to $N_B=1$ occurs.

Edit 2: It has been mentioned in the comments that the notion of critical exponents does not exist for finite system size as has been dealt with in the paper. My question is regarding the **infinite system size equivalent** of the case dealt with in the paper.

Edit 3: **Please note** that I'm not interested in spanning cluster percolation which is the usual percolation much spoken about. I'm interested in a different type of percolation i.e. when $N_B$ becomes $1$ and $MP \to LM$ occurs as has been discussed in the paper I linked in the first line. Please don't refer to the similar looking paper on ArXiv which isn't the same. It is talking about the general spanning cluster percolation which I'm not interested in.

wrongpaper which is talking aboutspanning clusterpercolation! $\endgroup$