  The interplay of topological constraints and persistence length of ring
polymers in their own melt is investigated by means of dynamical Monte Carlo
simulations of a three dimensional lattice model. We ask if the results are
consistent with an asymptotically regime where the rings behave like (compact)
{\em lattice animals} in a self-consistent network of topological constraints
imposed by neighbouring rings. Tuning the persistence length provides an
efficient route to increase the ring overlap required for this mean-field
picture to hold: The {\em effective} Flory exponent for the ring size decreases
down to $\nu \stackrel{<}{\sim} 1/3$ with increasing persistence length.
Evidence is provided for the emergence of one additional characteristic length
scale $\dtop \propto N^0$, only weakly dependent on the persistence length and
much larger than the excluded volume screening length $\xi$. At distances
larger than \dtop the conformational properties of the rings are governed by
the topological interactions, at smaller distances rings and their linear chain
counterparts become similar. (At distances smaller than $\xi$ both
architectures are identical.) However, the crossover between both limits is
intricate and broad as a detailed discussion of the local fractal dimension
(e.g., obtained from the static structure factor) reveals.
