We study the properties of ring polymers in disordered systems using a
Monte Carlo algorithm. The algorithm is used to generate a ring on a
two dimensional lattice, and the disorder is represented by the random
dilution of the lattice. We show how the ring undergoes a cross-over
from obeying self avoiding statistics at low concentrations of disorde
r, to behaving like a branched polymer as the concentration of disorde
r is increased. We find a scaling behavior to characterize this cross-
over phenomenon. We further show how this scaling behavior is also pre
sent in another class of problems, namely two dimensional vesicles sub
jected to a pressure differential.