We consider the fine-scale behavior of random walks on R^2 generated by successively applying independent random isometries to the origin.

We show that the walk equidistributes at a superpolynomial rate if the rotation parts satisfy a Diophantine condition,  and at a stretched exponential rate if the rotation parts are suitably algebraic. This extends results due to Varju and Lindenstrauss forisometries of R^3 to the more commutative and amenable setting of planar isometries. Based on joint work with Felipe Hernandez.