Dynamic site percolation of independent random walkers on 2-dimensional square lattice

Question

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.

Answer 1

If I understood correctly your question, you are looking to study the probability that a point $x$ is connected to the origin (for example) at time $t$. The probability of being connected depends on the distribution of your walkers.

Since the process of the walkers is ergodic, after a long time, your walkers will be distributed according to the reversible measure. Once you have identified the reversible measure, probably some kind of Poisson law conditionned to have $(2N+1)^2$ particles distributed over your $(2N+1)^2$ sites, then you can study the percolation function.

The distance between the distribution of your walkers and the reversible measure can be quantified using the spectral gap of the process, which should be of order $1/N^2$. Denote $P_t$ the law of the configuration of your walkers starting from your initial configuration and $\mu$ the reversible measure, then you should have: $$||P_t-\mu||_{TV}\leqslant C_1\times \exp(-C_2t/N^2)$$

I hope that this comment helped you.

About the spectral gap, I recommend the book of D. Levin, Y. Peres, and E. Wilmer: Markov chains and mixing times, that can be found at this address : http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf

Another book that can be useful is the book of Thomas Liggett: Interacting particle systems