In a square matrix if sites are independently set to be open with probability p (and therefore blocked with probability 1 − p), what is the probability that the system percolates?
A percolation model using an n-by-n grid of sites. Each site is either open or blocked. The system percolates if there is a connection of open sites possible from top edge to bottom. Each site can be independently open with a probability p. There is a threshold probability p* such that p above this would almost always percolate otherwise almost never percolates.
Here p* is estimated using Monte Carlo simulation, as follows:
- Initialize all sites to be blocked.
- Repeat the following unil system percolates:
- Choose a blocked site randomly
- Open the choosen site
- The fraction of sites that are open provides an estimate of the percolation threshold. Monte Carlo simulation takes t samples.
Here a 50x50 grid has been used and took 10,000 samples.
The file model.py has a class for the grid with all the required functions.
The file create-data.py creates the data using the model and creates data.csv.
notebook.ipynb is used to for analysis of data.csv.
Finally the threshold probability is found to be 0.59241956, with a standard deviation of 0.026125203.