Is the random walk on an isoradial graph defined in Chelkak and Smirnov (2011) (in pages 3-4) transient?

Let us define the random walk on an isoradial graph $\Gamma$ starting from $x$ by, \begin{equation} X_t = X_0 + \sum_{j=0}^{t-1} \xi_{X_j}^{(j)}, \end{equation} where $X_0 = x$ and the increments $\xi_{u}^{(t)}$ are independently and identically distributed for fixed vertex $u \in \Gamma$ and for all $t \in \mathbb{N}$. These are distributed according to, \begin{equation} \mathbb{P}(\xi_u = u_k - u) = \frac{\tan{\theta_k}}{\sum_{s=1}^{n}\tan{\theta_s}}, \end{equation} for $u_k$ $\sim$ $u$ are adjacent vertices in $\Gamma$. The half-angle of the rhombus is denoted by $\theta_k$. These are uniformly bounded from $0$ and $\pi/2$.

I am hesitating whether $(X_t)$ is transient? For fixed $u_0 \in \Gamma$ and given the mesh size $\delta > 0$, the free Green's function is finite: $G(u_0,u_0) = \frac{1}{2\pi}(\log \delta - \gamma_{Euler} -\log 2) < \infty$. Does this imply that $(X_t)$ is transient? In the light of Pólya's theorem this feels counter-intuitive. Since increments of $(X_t)$ have zero expectations, that is $\mathbb{E}(\Re \xi_u) = 0 $, $\mathbb{E}(\Im \xi_u) = 0 $, and $\Gamma$ is embedded in $\mathbb{C} \simeq \mathbb{R}^2$, one may expect that $(X_t)$ is recurrent.

Thank you for your time.


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