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

We consider a weighted lattice $Z^d$ with conductance $\mu_e=|e|^{-\alpha}$. 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 $\alpha\in (-1,0)$, two independent random walks on such weighted lattice will collide infinite many times while they are transient.