Critical Exponents for Island Mainland Transition (Percolation Theory)

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.

• It's not clear what you mean by $\chi$, $p_{c1}$ or $p_{c2}$ (you might include the definition of these so people looking at your question don't have to look at the papers). Jan 13, 2018 at 17:19
– user119567
Jan 13, 2018 at 17:36
• the "Island-Mainland" transition from arXiv:1612.04522 is actually a crossover, rather than a phase transition, without a diverging correlation length; this seems to preclude the notion of a critical exponent; in an infinite system the transition becomes the usual percolation transition, and then you know the critical exponents, but not for the finite system. Jan 15, 2018 at 16:43
• I don't think it is equivalent to the usual percolation transition which involves formation of a spanning black cluster. The percolation I'm interested is when $N_B$ becomes $1$ and it starts to look like white lakes in a black mainland. And that is what $p_{c2}$ represents in the paper I linked. It seems you have been reading the wrong paper which is talking about spanning cluster percolation!
– user119567
Jan 16, 2018 at 16:50

The problem considered in the Island-Mainland paper 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 and arXiv: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, 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.)

• "wrapping around a specific axis" just means that the connected cluster extends either from $x=0$ to $x=L$ (horizontal axis, denoted by $R_L^h$ in the paper) or from $y=0$ to $y=L$ (vertical axis, denoted by $R_L^v$); for a square lattice these wrapping probabilities are the same, but for other lattices they may differ. Jan 16, 2018 at 7:27
• Thanks for that definition. On re-reading your answer I have some confusions. You state $p_c=0.592746$ is the critical concentration (which is same whether we consider nearest neighbors or next nearest neighbors). BUT, in the paper I mentioned there are two critical probabilities $p_{c1}=0.13 \pm 0.005$ and $p_{c2}=0.87 \pm 0.005$.
– user119567
Jan 16, 2018 at 16:39
• According to their convention, $p_{c2}$ is actually the probability when the system percolates (MP $\to$ LM) i.e. a giant connected black cluster is formed such that $N_B=1$ (and not just a black spanning cluster which you seem to talk about at $p_c=0.592746$). BTW the paper you read and the paper I linked are not the same.
– user119567
Jan 16, 2018 at 16:39
• So basically, I don't find the value of $0.87 \pm 0.005$ anywhere on the net for a square infinite lattice. It's not on Wikipedia's Percolation Threshold page. In short your claim that the Island Mainland paper is just a "modification of conventional percolation problem" doesn't seem to be true.
– user119567
Jan 16, 2018 at 16:45
• Given that the bounty period is going to expire soon, would you please kindly reply to my query?
– user119567
Jan 20, 2018 at 7:00