Yes. <Simple random walk in two dimensions is recurrent>, so a walk started at $x_1$ hits $a$ almost surely and its loop erasure is a finite path. Exhaust $\mathbb Z^2$ by finite boxes, or use increasingly large tori with the marked vertices kept fixed. The probability that either walk reaches the boundary before hitting its target tends to zero by recurrence. The finite-graph reversal identity from part 2 therefore passes to the limit:
$$
\operatorname{LE}_{\mathbb Z^2}(x_1\to a)^{\leftarrow}
\overset d=
\operatorname{LE}_{\mathbb Z^2}(a\to x_1).
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 4 / 3 / Solution

A simple random walk on $\mathbb Z^2$ visits its starting point infinitely often almost surely and hits every fixed vertex almost surely.


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