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:

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:

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.

**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.