  We study reversible polymerization of rings. In this stochastic process, two
monomers bond and as a consequence, two disjoint rings may merge into a
compound ring, or, a single ring may split into two fragment rings. This
aggregation-fragmentation process exhibits a percolation transition with a
finite-ring phase in which all rings have microscopic length and a giant-ring
phase where macroscopic rings account for a finite fraction of the entire mass.
Interestingly, while the total mass of the giant rings is a deterministic
quantity, their total number and their sizes are stochastic quantities. The
size distribution of the macroscopic rings is universal, although the span of
this distribution increases with time. Moreover, the average number of giant
rings scales logarithmically with system size. We introduce a card-shuffling
algorithm for efficient simulation of the ring formation process, and present
numerical verification of the theoretical predictions.
