# Is there a transient graph whose spectral dimension two?

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 $$x \in V(G)$$.

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.