I was looking at this paper [“Islands in Sea” and “Lakes in Mainland” phases and related transitions simulated on a square lattice](https://drive.google.com/file/d/1onBqzq50ApMzyvgMYekvjXdUXRbuhoZN/view?usp=sharing) 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](https://en.wikipedia.org/wiki/Critical_exponent) 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](https://arxiv.org/abs/1612.04522) which isn't the same. It is talking about the general spanning cluster percolation which I'm not interested in.



Critical Exponents for Island Mainland Transition (Percolation Theory)
The problem considered in the [Island-Mainland paper](https://arxiv.org/abs/1612.04522) is site-percolation on a two-dimensional square lattice, with one modification of the conventional problem: two squares of the same colour are considered connected if they are nearest-neighbor (they share an edge) or next-nearest-neighbor (they share a vertex). This modification has no effect on the percolation transition, see [arXiv:cond-mat/0408338](https://arxiv.org/pdf/cond-mat/0408338.pdf)  and [arXiv:cond-mat/0609635](https://arxiv.org/abs/cond-mat/0609635): the critical concentration $p\_c=0.592746$ is the same whether we only consider nearest-neighbors or include next-nearest-neighbors in a connected cluster.


So the "mixed phase" in the Island-Mainland paper, where neither the black nor white squares are fully connected, occurs for concentrations $p$ in the interval $(0.41,0.59)$. This is for an infinite lattice. The cited paper studies this numerically for a $50\times 50$ lattice, and finds for the mixed phase an interval $(0.39,0.61)$, which is already quite close to the infinite-lattice limit.


Concerning power law scaling and critical exponents, see [arXiv:cond-mat/0304024](https://arxiv.org/abs/cond-mat/0304024), where the "wrapping probability" $R\_L$ for site percolation on an $L\times L$ square lattice with periodic boundary conditions was calculated. (This is the probability that a cluster grows large enough to wrap around in the horizontal direction.) The derivative $dR\_L/dp$ at the critical concentration scales as
$$\frac{dR\_L}{dp}\propto L^{1/\nu}$$
with critical exponent $\nu=4/3$. This is expected to be a universal result, the same for nearest-neighbor and next-nearest-neighbor connectivity, so it should apply to the Island-Mainland paper.


The plot below, from arXiv:cond-mat/0304024, shows the power law scaling of $dR\_L/dp$ fitted to $\nu=1.334$ for the square lattice. (The exponent is different for a hexagonal lattice.)
![](https://ilorentz.org/beenakker/MO/percolation.png)


