  The hull perimeter at distance d in a planar map with two marked vertices at
distance k from each other is the length of the closed curve separating these
two vertices and lying at distance d from the first one (d<k). We study the
statistics of hull perimeters in large random planar triangulations and
quadrangulations as a function of both k and d. Explicit expressions for the
probability density of the hull perimeter at distance d, as well as for the
joint probability density of hull perimeters at distances d1 and d2, are
obtained in the limit of infinitely large k. We also consider the situation
where the distance d at which the hull perimeter is measured corresponds to a
finite fraction of k. The various laws that we obtain are identical for
triangulations and for quadrangulations, up to a global rescaling. Our approach
is based on recursion relations recently introduced by the author which
determine the generating functions of so-called slices, i.e. pieces of maps
with appropriate distance constraints. It is indeed shown that the map
decompositions underlying these recursion relations are intimately linked to
the notion of hull perimeters and provide a simple way to fully control them.
