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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 atwo-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 weightedlattice will collide infinite many times while they are transient.

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