  Let $(M,d,\mu)$ be a uniformly discrete metric measure space satisfying space
homogeneous volume doubling condition. We consider discrete time Markov chains
on $M$ symmetric with respect to $\mu$ and whose one-step transition density is
comparable to $ (V_h(d(x,y)) \phi(d(x,y))^{-1}$, where $\phi$ is a positive
continuous regularly varying function with index $\beta \in (0,2)$ and $V_h$ is
the homogeneous volume growth function. Extending several existing work by
other authors, we prove global upper and lower bounds for $n$-step transition
probability density that are sharp up to constants.
