Random Walk in 2D

Probability Level 3

Consider a random walk along the lattice points of the 2D coordinate plane, starting at the origin. Each step, the walk progresses randomly to an adjacent point in any of the four cardinal directions (positive yy, negative yy, positive xx, negative xx), and this process continues forever.

Will the walk inevitably (i.e., with probability 1) return to the origin at some point in time? What about in 1D?

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