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We study the thermodynamical properties of black holes when described asgases of indistinguishable punctures with a chemical potential. In thispicture, which arises from loop quantum gravity, the black hole microstates aredefined by finite families of half-integers spins coloring the punctures, andthe near-horizon energy measured by quasi-local stationary observers definesthe various thermodynamical ensembles. The punctures carry excitations ofquantum geometry in the form of quanta of area, and the total horizon area$a_\text{H}$ is given by the sum of these microscopic contributions. We assumehere that the system satisfies the Bose-Einstein statistics, and that eachmicrostate is degenerate with a holographic degeneracy given by$\exp\big(\lambda a_\text{H}/\ell_\text{Pl}^2\big)$ and $\lambda>0$. We analyzein detail the thermodynamical properties resulting from these inputs, and inparticular compute the grand canonical entropy. We explain why the requirementsthat the temperature be fixed to the Unruh temperature and that the chemicalpotential vanishes do not specify completely the semi-classical regime of largehorizon area, and classify in turn what the various regimes can be. When thedegeneracy saturates the holographic bound ($\lambda=1/4$), there exists asemi-classical regime in which the subleading corrections to the entropy arelogarithmic. Furthermore, this regime corresponds to a Bose-Einsteincondensation, in the sense that it is dominated by punctures carrying theminimal (or ground state) spin value $1/2$.

Black holes as gases of punctures with a chemical potential: Bose-Einstein condensation and logarithmic corrections to the entropy