Consider a nearest-neighbor (or say simple) random walk on a connected graph $G$ with infinite vertices where each vertex has a finite degree. Let $P^n_{o,o}$ be the probability of the random walk starting at vertex $o$ and returning to $o$ after $n$ steps. We write $a_n\sim b_n$ to mean $\lim_n a_n/b_n=1$.
It is a classical result that for a simple random walk on $\mathbb{Z}$, we have $P^{2n}_{o,o}\sim c n^{-1/2}$ as $n\rightarrow\infty$ for a constant $c>0$.
My question is, are there examples of graph $G$, where the relation $$P^{n}_{o,o}\sim c n^{-\alpha}$$ (or $P^{2n}_{o,o}$ if $G$ is bipartite) is precisely obtained for $\alpha$ in the full range of the interval (0,1)?
PS:
(1) I need this range of $\alpha$ to ensure that the walk is recurrent.
(2) In the monograph of Woess (2000), I have only been able to find this relation for countable $\alpha$'s within $(0,1)$. For example, for the $d$-dimensional comb lattice, the relation holds with $\alpha= 2^{-d}-1$ (Proposition 18.4 of Woess (2000)).
