Let $P_{x,y}$ the probability that a random walk starting from $x$ will ever visit $y$. Consider a biased random walk in $\mathbb{Z}^2$. Let $A_k$ be the set of vertices having a distance less than $k$ from the origin. Let $$C_k := \sum\limits_{x \in A_k} P_{x,0}.$$
Is it possible to determine an explicit asymptotic formula for $C_k$ as $k$ is large? For example, does it grow linearly with $k$?