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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 $x \in V(G)$.

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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

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