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Correlated stability conjecture revisited

Correlated stability conjecture (CSC) proposed by Gubser and Mitra [1,2] linked the thermodynamic and classical (in)stabilities of black branes. The classical instabilities, whenever occurring, were conjectured to arise as Gregory-Laflamme (GL) instabilities of translationally invariant horizons. 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. A counter-example of CSC was presented in the context of black branes with scalar hair undergoing a second-order phase transition [4]. In this paper we discuss a related counter-example of CSC conjecture, where a thermodynamically stable translationally invariant horizon has a genuine tachyonic instability. We study the spectrum of quasinormal excitations of a black brane undergoing a continuous phase transition, and explicitly identify the instability. We compute the critical exponents of the critical momenta and the frequency of the unstable fluctuations and identify the dynamical critical exponent of the model.