Let $G = (V(G), E(G))$ be an infinite connected simple graph. 
Let $((S\_n)\_n, (P^x)\_{x \in V(G)})$ be the simple random walk on $G$. 
Let $p\_n (x,y) = P^x (S\_n = y)$. 
A spectral dimension of $G$ is given by the following:
$$ d(G) = -2\lim\_{n \to \infty} \frac{\log p\_{2n}(x,x)}{\log n}, x \in V(G)$$
if the limit exists. (If the limit exists then the value does not depend on $x$.)


If $d(G) > 2$, then $G$ is transient, specifically, the simple random walk on $G$ is transient. 
If $d(G) < 2$, then $G$ is recurrent, specifically, the simple random walk on $G$ is recurrent. 
It is well-known that if $G$ is the square lattice $\mathbb Z^2$, then $d(\mathbb Z^2) = 2$. 
I would like to know whether there exists an infinite transient connected simple graph whose spectral dimension exists and is equal to $2$. 
For example this situation occurs if there exist an infinite connected graph $G$ and two positive constants $c\_1, c\_2$ such that 
$$ c\_1 \le n (\log n)^2 p\_{2n}(x,x) \le c\_2$$ holds for every $n \ge 1$ and $x \in V(G)$. 



Is there a transient graph whose spectral dimension two?
Consider the wedge $\{(x,y,z) \in {\bf Z}^3 : |z| \le \log^2(|x|+2)\}$. It is transient by
[1] yet close enough to the two dimensional lattice so that it has spectral dimension 2.


[1] T.J Lyons : A Simple Criterion for Transience of a Reversible Markov Chain
<https://projecteuclid.org/euclid.aop/1176993604>


