The phase transition between caged black holes and black strings

Black-hole uniqueness is known to fail in higher dimensions, and the multiplicity of black hole phases leads to phase transitions physics in General Relativity. The black-hole black-string transition is a prime realization of such a system and its phase diagram has been the subject of considerable study in the last few years. The most surprising results seem to be the appearance of critical dimensions where the qualitative behavior of the system changes, and a novel kind of topology change. Recently, a full phase diagram was determined numerically, confirming earlier predictions for a merger of the black-hole and black-string phases and giving very strong evidence that the end-state of the Gregory-Laflamme instability is a black hole (in the dimension range 5 ≤ D ≤ 13). Here this progress is reviewed, illustrated with figures, put into a wider context, and the still open questions are listed.