  We analyze several families of two-dimensional quantum random walks. The
feasible region (the region where probabilities do not decay exponentially with
time) grows linearly with time, as is the case with one-dimensional QRW. The
limiting shape of the feasible region is, however, quite different. The limit
region turns out to be an algebraic set, which we characterize as the rational
image of a compact algebraic variety. We also compute the probability profile
within the limit region, which is essentially a negative power of the Gaussian
curvature of the same algebraic variety. Our methods are based on analysis of
the space-time generating function, following the methods of Pemantle and
Wilson (2002).
