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Multi-black rings and the phase diagram of higher-dimensional black holes

Configurations of multiple concentric black rings play an important role in determining the pattern of branchings, connections and mergers between different phases of higher-dimensional black holes. We examine them using both approximate and (in five dimensions) exact methods. By identifying the role of the different scales in the system, we argue that it is possible to have multiple black ring configurations in which all the rings have equal temperature and angular velocity. This allows us to correct and improve in a simple, natural manner, an earlier proposal for the phase diagram of singly-rotating black holes in $D\geq 6$.