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We consider a boson gas on the stretched horizon of the Schwartzschild andKerr black holes. It is shown that the gas is in a Bose-Einstein condensedstate with the Hawking temperature $T_c=T_H$ if the particle number of thesystem be equal to the number of quantum bits of space-time $ N \simeq{A}/{{\l_{p}}^{2}}$. Entropy of the gas is proportional to the area of thehorizon $(A)$ by construction. For a more realistic model of quantum degrees offreedom on the horizon, we should presumably consider interacting bosons(gravitons). An ideal gas with intermediate statistics could be considered asan effective theory for interacting bosons. This analysis shows that we mayobtain a correct entropy just by a suitable choice of parameter in theintermediate statistics.

Condensation of an ideal gas with intermediate statistics on the horizon