  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.
