How Does Finite Size Scaling Reveal Cluster Behavior in 1D Percolation?

[Mikkel]

TL;DR Summary

Simulate 1d percolation. I have to show that the probability of any site belonging to the largest cluster vanishes as N -> infinity

Hello

I am struggeling with a problem, or perhaps more with understanding the problem.
I have to simulate a one dimensional percolation in Python and that part I can do. The issue is understanding the next line of the problem, which I will post here:
"For the largest cluster size S, use finite size scaling, i.e., allow N to increase and plot s ≡ S/N vs. 1/N, to show that the probability of any site to belong to the largest cluster vanishes in the thermodynamic limit. Hint: Use N raised to some power between 2 and 5".
So, the way I understand this is to, let N increase some amount each iteration and find the largest cluster. I save these values and plot S/N vs. 1/N ending up with the attached plot.
I'm just unsure wheter or not this is correctly interpreted and would love to hear others input

Thanks!

 

[Baluncore]

I think you need to post the original question, exactly as it was presented.
If you present your interpretation only, it will prevent us from identifying your misunderstanding.
Please provide a reference, or a link to the original source of the question.

 

[Mikkel]

I think you need to post the original question, exactly as it was presented.
If you present your interpretation only, it will prevent us from identifying your misunderstanding.
Please provide a reference, or a link to the original source of the question.

You might be right, but the whole question is pretty much in the quotation marks. I will post the entire question below:
"For a given value of p, 0 ≤ p ≤ 1, numerically find the largest connected cluster of sites. For the largest cluster size S, use finite size scaling, i.e., allow N to increase and plot s ≡ S/N vs. 1/N, to show that the probability of any site to belong to the largest cluster vanishes in the thermodynamic limit. (Hint: reasonable system sizes for finite size scaling are N = 10^m, with m ∈ {2,3,4,5}.)"

 

[Baluncore]

Maybe this helps.
https://sandipanweb.wordpress.com/2...on-threshold-via-monte-carlo-simulation-in-r/

 

[Mikkel]

Maybe this helps.
https://sandipanweb.wordpress.com/2...on-threshold-via-monte-carlo-simulation-in-r/

Thank you for your input.
This is similar to what I've found elsewhere. I can show a lot of interesting effects in my 2D model, but nothing "special" happens in 1D, which is what confuses me, because I'm not sure what to except.

 

[Baluncore]

3D percolation involves connectivity from the top 2D layer to the bottom 2D layer.
2D percolation involves connectivity from the top row of sites to the bottom row of sites.
1D percolation is more confusing.

You could consider a 1D column of sites. It will be open only if all sites from top to bottom are open.
Alternatively, a horizontal 1D line of sites, with percolation downwards, across one layer only, will be open if anyone site is open.

How do you visualise a 1D array of sites ?

http://web.mit.edu/8.334/www/grades/projects/projects10/Gardner_Webpage/OneD.htm