Note on the thermodynamic stability of a black ring at quantum scales

In this paper, the thermodynamic properties of a thin black ring in AdS space-time is explored when the size of the ring is comparable to quantum scales. The angular momentum to mass ratio of this system has an upper limit, which is the cosmological radius of the black ring. It is found that the small black ring will be thermodynamically stable due to the effects introduced by thermal fluctuations. However, we find that the black ring is less stable than thermal AdS. Thermodynamic analysis indicates that there is no critical point, but there is Hawking-Page transition to radiation, which is confirmed by the Gibbs free energy analysis.