Let $p_{t,v}$ be the probability a discrete time, simple random walk at $v \in \mathbb Z^2$ reaches 0 within $t$ steps. We are looking for a reference to cite or an easy proof for the big-O behavior of $p_{t,v}$ with respect to $\|v\|$. We don't care about the constant.
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$\begingroup$ Isn't it the local CLT? In any case, both Spitzer and Lawler's books will have plenty of material on this including very sharp asymptotics of the form $C/t(1+o(1))$ with explicit $C$ $\endgroup$– ofer zeitouniCommented Aug 21, 2017 at 6:37
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$\begingroup$ Yes. This is local CLT. Thanks. Lawler Chapter 2 appears to have what we need. I haven't worked it out yet, but suspect $p_{t,v}$ will behave like $C/(\|v\| \log \|v\|)$. This is on account of it spending $\log t$ time revisiting sites. I'll post when I do the calculation. $\endgroup$– Matthew JungeCommented Aug 21, 2017 at 14:31
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