Let $G$ be a bounded degree graph and fix a vertex $v_0$. Suppose that the simple random walk on $G$ is transient, and let $g:G\to\mathbb{R}$ be defined by $$g(v)=\mathbb{P}_v[T_{v_0}<\infty].$$ That is, $g(v)$ is the probability that a simple random walk starting at $v$ will ever hit $v_0$.

Now consider $g(G)$, the image of $g$ in $[0,1]$.

Can $g(G)$ have an accumulation point other than $0$?