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The Virasoro Gibbs state and BTZ black holes

We show that the Virasoro Gibbs state accurately describes the thermodynamics of BTZ black holes at large temperatures and in the strong-coupling regime $c<1$. We first give a simple heuristic argument by showing that at high temperatures and arbitrary $c$, the energies are much larger than Planck mass. Then we give a detailed analysis of the quantum fluctuations of the Gibbs state on unitary irreducible representations of Virasoro group and explicitly show that they go to zero in the high-temperature limit by means of representation theory results. This implies the state has a sensible thermodynamic limit which actually holds for any $c$. Finally, the matching with BTZ thermodynamics for $c<1$ is obtained by using the known asymptotic behaviour of the characters of the Virasoro discrete series. This result supports the idea that minimal models could describe quantum gravity at strong coupling. We make no use of the Euclidean path integral nor assume modular invariance of the partition functions, although we comment on them along the way.