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I apologize for the title gore. I was recently studying percolation theory to work on improving the theoretical work behind some research I'm doing with a professor. While working on it, a thought came to my head that I needed answering but couldn't find an answer to.

In one of the classical percolation problems, we seek to find the critical probability $p_c$ that a percolating cluster exists on a graph. That is, if $p > p_c$, a cluster exists, and for $p < p_c$, it does not. With much effort, it was found that for the case on the 2d lattice, $p_c = 0.5$.

Similarly, on the random walk on a d-dimensional graph, it was proved that for $p = \frac{1}{2d}$, the DTMC is recurrent, but for any other probability will be transient. That is, the expected amount of times one will return to a previous location as finite.

When I envision these two different cases in my head, I feel that there are some similarities, but I could be completely off base. Can there be found any relation between the two problems?

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I think that the most we can say is that we expect that percolation at criticality on transitive graphs (to be more precise at unicity of the infinite cluster but on $\mathbb{Z}^d$ it's the same) should behave as a branching random walk at criticality.

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