Quantum random walks on integer lattices

Motivated by quantum information theory, a number of
nearest neighbor quantum particle models have been
introduced, mostly in the last five years, under the
name quantum random walk (QRW). Many of these results
may be unified and extended by working directly with
a space-time generating function. Obtaining the
generating function is in fact quite simple.

Transferring this back to limiting behavior of QRW is
a matter of harnessing recent technology concerning
asymptotics of multivariate generating functions.
The accompanying picture shows the probability profile
of a particular two-dimensional quantum random walk after
100 steps. I hope to show how you can guess this picture
from a glance at the generating function.