[apetersson]

Doesn't that only happen in certain dimensionality < 3 ? See also random walks in higher dimensions

[isotropy]

You may be thinking of Polya's result, which lets the walker go anywhere on the grid, so there are more possible states than Poincaré allows. For a discrete system, Poincaré assumes (roughly) that the space of possibilities is finite.

[red_trumpet]

Poincaré recurrence is not about random walks though.