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Spherical perturbations of hairy black holes in designer gravity theories

We study the spectrum of scalar $l=0$ quasi-normal frequencies of Anti-de-Sitter hairy black holes in four and five dimensional designer gravity theories of the Einstein-scalar type, arising as consistent truncations of $ \mathcal{N} = 8$ gauged supergravity. In the dual field theory, such hairy black holes represent thermal states in which the operator corresponding to the bulk scalar field is condensed, due to the multi-trace deformation associated with non-standard boundary conditions. We show that, in a particular class of models, the effective potential describing the vacua of the deformed dual theory can be identified, at large values of the condensate, with the deformation plus the conformal coupling of the condensate to the curvature of the boundary geometry. In this limit, we show that the least damped quasi-normal frequency of the corresponding hairy black holes can be accurately predicted by the curvature of the effective potential describing the field theory at finite entropy.