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

We consider a weighted lattice $Z^d$ with conductance ${\mu }_e=\mathrm{\mid }e{\mathrm{\mid }}^{-\alpha }$. 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.