  The conformational statistics of ring polymers in melts or dense solutions is
strongly affected by their quenched microscopic topological state. The effect
is particularly strong for untangled (i.e. non-concatenated and unknotted)
rings, which are known to crumple and segregate. Here we study these systems
using a computationally efficient multi-scale approach, where we combine
massive simulations on the fiber level with the explicit construction of
untangled ring melt configurations based on theoretical ideas for their large
scale structure. We find (i) that topological constraints may be neglected on
scales below the standard entanglement length, $L_e$, (ii) that rings with a
size $1 \le L_r/L_e \le 30$ exhibit nearly ideal lattice tree behavior
characterized by primitive paths which are randomly branched on the
entanglement scale, and (iii) that larger rings are compact with gyration radii
$\langle R_g^2(L_r) \rangle \propto L_r^{2/3}$. The detailed comparison between
equilibrated and constructed ensembles allows us to perform a `Feynman test' of
our understanding of untangled rings: can we convert ideas for the large-scale
ring structure into algorithms for constructing (nearly) equilibrated ring melt
samples? We show that most structural observables are quantitatively reproduced
by two different construction schemes: hierarchical crumpling and ring melts
derived from the analogy to interacting branched polymers. However, the latter
fail the `Feynman test' with respect to the magnetic radius, $R_m$, which we
have defined based on an analogy to magnetostatics. While $R_m$ is expected to
vanish for double-folded structures, the observed values of $\langle R_m^2(L_r)
\rangle \propto \langle R_g^2(L_r) \rangle$ provide a simple and
computationally convenient measure of the presence of a non-negligible amount
of local loop opening in crumpled rings.
