Dynamic site percolation of independant random walkers on 2d square lattice Really appreciate your help, I am stuck in a part of my research which I am not expert in.
I have a 2-dimensional square lattice with periodic boundary conditions(torus). I am placing one walker at each node at the beginning. Then I let the walkers do a lazy random walk, which means that with probability 1/2 they jump to one of their 4 nearest neighbors with the same probability in each time-step. Walkers are independent so they can occupy the same site.
I am looking at this problem as a site percolation problem, so if two neighboring sites have at least one walker on them they belong to the same cluster.
At t=0, the particle density on each of my sites is 1, so I am at the endpoint of percolation probability function. But as the walkers do their first jump the percolation probability will change because the density of number of walkers per site is less than one in each timestep.  
I am looking for the time behavior of percolation probability function in this problem.
Any idea or relevant paper would be very helpful.
Thank You
Klara
 A: I am not sure if 
$\textbf{1)}$ you have two particle that are randomly walking and required to fit on a lattice (like checkers)
or
$\textbf{2)}$ you have two particle which will split up so that a fraction of its density goes into the surrounding cells (like a diffusion process). 
$\textbf{Case 1}$ 
First the probability that the particle will jump to one if its nearest neighbors should be 1/4 not 1/2, so that the probability is equal for each direction.
Writing a code to preform this monte-carlo should not be terribly difficult to do a brute force calculation. Essentially you would need to create a NxN matrix of integers. A particle would then be identified by the coordinate say (x,y). On each step each particle has four possible end states given by
$\begin{matrix}
1 & (x+1,y) \\
2 & (x-1,y) \\
3 & (x,y+1) \\
4 & (x,y-1)
\end{matrix}$
You could then randomly draw an integer from 1:4. The result of the random draw would then be the new coordinate of the particle. You would then preform the same for your second particle. The tricky part would be the periodic boundary conditions, but after some thinking you could figure it out using a modular function.
$\textbf{Case 2}$
I have done some research on a related topic called variance reduction techniques. This arrises when you try to preform case one, but you need way more data than any computer could hand. So we use our some math to make the calculation more efficient. The idea is very similar to case one.
You preform the same transport process I described above but in one dimension (much simpler). But to retain the two dimensional nature of the problem instead of allowing the density in the new cell to be one, you make it some fraction of one. This simulates the effect that some fraction of the particle has left into the second dimension. 
Typically you randomly sample from some posterior probability distribution function to determine how much density you let through.
I would do a search of variance reduction techniques, Metropolis algorithm, and you might find chapter two helpful here (http://dspace.nwu.ac.za/bitstream/handle/10394/3841/vanderwaltdekock_marisa.pdf?sequence=1)
Hope this helps you search!
A: Thank you for your response. I have N walkers where N is the number of nodes of my lattice. At the beginning, all of the nodes have one walker on them. I have written the code to simulate this. I am looking for analytical understanding of the problem.
