Stability and Critical Phenomena of Black Holes and Black Rings
  We revisit the general topic of thermodynamical stability and critical phenomena in black hole physics, analyzing in detail the phase diagram of the five dimensional rotating black hole and the black rings discovered by Emparan and Reall. First we address the issue of microcanonical stability of these spacetimes and its relation to thermodynamics by using the so-called Poincare (or "turning point") method, which we review in detail. We are able to prove that one of the black ring branches is always locally unstable, showing that there is a change of stability at the point where the two black ring branches meet. Next we study divergence of fluctuations, the geometry of the thermodynamic state space (Ruppeiner geometry) and compute the appropriate critical exponents and verify the scaling laws familiar from RG theory in statistical mechanics. We find that, at extremality, the behaviour of the system is formally very similar to a second order phase transition. 
Giovanni Arcioni and Ernesto Lozano-Tellechea