Gaussian bounds and collisions of variable speed random walks on lattices with power law conductances

We consider a weighted lattice Zd with conductance μe=∣e∣−α. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when d=2 and α∈(−1,0), two independent random walks on such weighted lattice will collide infinitely many times while they are transient.