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. It means that the number of walkers is equal to the number of sites of my lattice. Then I let the walkers do a lazy random walk, which means that with probability 1/2 they don't move at all, and if they move, they jump to one of their 4 nearest neighbors with the same probability(1/4) 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.