  The distribution of monomers in a coating of grafted and adsorbing polymers
is modelled using a grafted staircase polygon in the square lattice. The
adsorbing staircase polygon consists of a bottom and a top lattice path
(branches) and the asymptotic probability density that vertices in the lattice
are occupied by these paths is determined. This is a model consisting of two
linear polymers grafted to a hard wall in a coating. The probability that
either the bottom, or the top, path of a staircase polygon passes through a
lattice site with coordinates $(\lfloor \epsilon n \rfloor,\lfloor \delta
\sqrt{n}\rfloor)$, for $0 < \epsilon < 1$ and $\delta\geq 0$, is determined
asymptotically as $n\to\infty$. This gives the probability density of vertices
in the staircase polygon in the scaling limit (as $n\to\infty$):
  $\displaystyle {\mathbb P}_r^{f,(both)} (\epsilon,\delta) =
\frac{\delta^2(15\,\epsilon^2(1 - \epsilon)^2 - 12\, \delta^2 \epsilon(1 -
\epsilon) + 4\, \delta^4 ) }{3\sqrt{\pi\,\epsilon^7(1 - \epsilon)^7}} \,
e^{-\delta^2/\epsilon(1 - \epsilon)} .$
  The densities of other cases, grafted and adsorbed, and at the adsorption
critical point (or special point), are also determined. In these cases the most
likely and mean trajectories of the paths are determined. The results show that
low density regions in the distribution induces entropic forces on test
particles which confine them next to the hard wall, or between branches of the
staircase polygon. These results give qualitative mechanisms for the
stabilisation of a drug particle confined to a polymer coating in a drug
delivery system such as a drug-eluding stent covered by a grafted polymer.
