Melt of polymer rings: The decorated loop model

Melts of unconcatenated and unknotted polymer rings are a paradigm for soft matter ruled by topological interactions. We propose a description of a system of rings of length N as a collection of smaller polydisperse Gaussian loops, ranging from the entanglement length to the skeleton ring length , assembled in random trees. Individual rings in the melt are predicted to be marginally compact with a mean square radius of gyration . As a rule, simple power laws for asymptotically long rings come with sluggish crossovers. Experiments and computer simulations merely deal with crossover regimes typically extending to . The estimated crossover functions allow for a satisfactory fit of simulation data.