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Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "white"). If any cell, lies in the Von Neumann neighbourhood of a certain cell and also has the same colour as of that central cell, it is said to belong to the same cluster as that central cell. Moreover, if any cell belongs to the Moore neighbourhood of a certain cell, but not its Von Neumann neighbourhood, and is of the same as of that cell, it is considered to belong the cluster as of that central cell with a probability $q$.

I wrote a program to plot the "Euler number" graphs, that is, the $\chi(p) \ [=N_B(p)-N_W(p)]$ vs. $p$ graphs, for different values of $q$, where $N_B(p)$ is the number of black clusters and $N_W(p)$ is the number of white clusters, at a probability $p$.

For a $1000\times 1000$ matrix (averaged over $100$ iterations) the Euler number graph's variation with $q$ is as follows:

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When $q=0.5$ the middle root of the curve is extremely close to $0.5$.

I plotted the middle roots ($p_0$'s) in another graph:

enter image description here

For $1000\times 1000$ the middle root $p_0$'s variation with $q$ seems to be almost linear. Also, I plotted the same graph for a few more sizes: $125\times 125$, $250\times 250$ and $500\times 500$. I noticed that as system size increases the "middle root" graph gets more and more smooth and linear.

For what it's worth, I also noticed a similar trend (i.e. "site percolation threshold vs. $q$" graphs getting linear and smoother with increasing size) for the (approximate) site percolation thresholds for these finite size lattices.

enter image description here

Is there any mathematical justification for this trend?


P.S: Answers addressing only the site percolation threshold trend or only the $p_0$ trend are also welcome.

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