Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously visited (these coordinates become fully reflecting to the particle). As such, the particle may trap itself if its trajectory generates a closed loop.

Assuming the Brownian particle has some diffusion coefficient, $D$, is there a known mean and variance for the maximum distance the particle will travel from its origin before trapping itself?

(Does anyone have a reference for the maximum distance from origin mean/variance for a random walker on an infinite two-dimensional integer lattice?)