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The conformational statistics of ring polymers in melts or dense solutions isstrongly affected by their quenched microscopic topological state. The effectis particularly strong for untangled (i.e. non-concatenated and unknotted)rings, which are known to crumple and segregate. Here we study these systemsusing a computationally efficient multi-scale approach, where we combinemassive simulations on the fiber level with the explicit construction ofuntangled ring melt configurations based on theoretical ideas for their largescale structure. We find (i) that topological constraints may be neglected onscales below the standard entanglement length, $L_e$, (ii) that rings with asize $1 \le L_r/L_e \le 30$ exhibit nearly ideal lattice tree behaviorcharacterized by primitive paths which are randomly branched on theentanglement 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 betweenequilibrated and constructed ensembles allows us to perform a `Feynman test' ofour understanding of untangled rings: can we convert ideas for the large-scalering structure into algorithms for constructing (nearly) equilibrated ring meltsamples? We show that most structural observables are quantitatively reproducedby two different construction schemes: hierarchical crumpling and ring meltsderived from the analogy to interacting branched polymers. However, the latterfail the `Feynman test' with respect to the magnetic radius, $R_m$, which wehave defined based on an analogy to magnetostatics. While $R_m$ is expected tovanish 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 andcomputationally convenient measure of the presence of a non-negligible amountof local loop opening in crumpled rings.

Local loop opening in untangled ring polymer melts: A detailed "Feynman test" of models for the large scale structure