Consider a random walk in $\mathbb{Z}^2$, $x(j) = x(j-1) + \xi_j$, where the increments are random variables independent and identically distributed with finite support, the expectation $m := \mathbb{E}[\xi]$ is a vector that lies entirely on the horizontal axes $e_1$ and $x(0) = 0$.
Call $\mathcal{C} \subset \mathbb{R}^2$ the line orthogonal to $e_1$ and intersecting the point $(t,0)$.
Call $p(t, i)$ the probability that the walker will intersect for the first time the line $\mathcal{C}$ on the site $(t,i)$, where $t \in \mathbb{N}$ and $i \in \mathbb{Z}$.
How does $p(t, i)$ scales with $t$? Is it true that for all $i$ (or at least for some small $|i|$) $\lim_{t \rightarrow \infty} p(t,i) / t = c_i$, where $c_i$ is a positive constant? Or should I rescale with $\sqrt(t)$? How to prove it?
In the simple random walk case, call $(t,Y_t)$ the hitting point of the line $te_1+\mathbb Ze_2$, then $Y_t/t$ converges in distribution to a standard Cauchy random variable hence $p(t,i)\sim1/(t\pi)$ for every fixed $i$, when $t\to+\infty$. This indicates that, in the general case, the proper renormalization is most probably $t$, not $\sqrt{t}$.