Arithmetic aspects of short random walks

We revisit a classical problem: how far does a random walk travel in a given
number of steps (of length 1, each taken along a uniformly random direction)?
Although such random walks are asymptotically well understood, surprisingly
little is known about the exact distribution of the distance after just a few
steps.  For instance, the average distance after two steps is (trivially)
given by 4/pi; but what is the average distance after three steps?

In this talk, we therefore focus on the arithmetic properties of short random
walks and consider both the moments of the distribution of these distances as
well as the corresponding density functions.  It turns out that the even
moments have a rich combinatorial structure which we exploit to obtain
analytic information.  In particular, we find that in the case of three and
four steps, the density functions can be put in hypergeometric form and may be
parametrized by modular functions.  Much less is known for the density in case
of five random steps, but using the modularity of the four-step case we are
able to deduce its exact behaviour near zero.

Time permitting, we will also discuss connections with Mahler measure.