= Solution
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:
$$
\operatorname{LE}_{\mathbb Z^2}(x_1\to a)^{\leftarrow}
\overset d=
\operatorname{LE}_{\mathbb Z^2}(a\to x_1).
$$
Back to article page
