  We study the convex hull of the set of points visited by a two-dimensional
random walker of T discrete time steps. Two natural observables that
characterize the convex hull in two dimensions are its perimeter L and area A.
While the mean perimeter <L> and the mean area <A> have been studied before,
analytically and numerically, and exact results are known for large T (Brownian
motion limit), little is known about the full distributions P(A) and P(L). In
this paper, we provide numerical results for these distributions. We use a
sophisticated large-deviation approach that allows us to study the
distributions over a larger range of the support, where the probabilities P(A)
and P(L) are as small as 10^{-300}. We analyze (open) random walks as well as
(closed) Brownian bridges on the two-dimensional discrete grid as well as in
the two-dimensional plane. The resulting distributions exhibit, for large T, a
universal scaling behavior (independent of the details of the jump
distributions) as a function of A/T and L/\sqrt{T}, respectively. We are also
able to obtain the rate function, describing rare events at the tails of these
distributions, via a numerical extrapolation scheme and find a linear and
square dependence as a function of the rescaled perimeter and the rescaled
area, respectively.
