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Can the correlated stability conjecture be saved?

Correlated stability conjecture (CSC) proposed by Gubser and Mitra [1,2] linked the thermodynamic and classical (in)stabilities of black branes. In [3] it was shown that the thermodynamic instabilities, specifically the negative specific heat, indeed result in the instabilities in the hydrodynamic spectrum of holographically dual plasma excitations. Counter-examples of CSC were presented in the context of black branes with scalar hair undergoing a second-order phase transition [4,5]. The latter translationary invariant horizons have scalar hair, raising the question whether the asymptotic parameters of the scalar hair can be appropriately interpreted as additional charges leading to a generalization of the thermodynamic stability criterion. In this paper we show that the generalization of the thermodynamic stability criterion of this type can not save CSC. We further present a simple statistical model which makes it clear that thermodynamic and dynamical (in)stabilities generically are not correlated.