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Hair on near-extremal Reissner-Nordstrom AdS black holes

We discuss hairy black hole solutions with scalar hair of scaling dimension $Δ$ and (small) electromagnetic coupling $q^2$, near extremality. Using trial functions, we show that hair forms below a critical temperature $T_c$ in the region of parameter space $(Δ, q^2)$ above a critical line $q_c^2 (Δ)$. For $Δ> Δ_0$, the critical coupling $q_c^2$ is determined by the AdS$_2$ geometry of the horizon. For $Δ< Δ_0$, $q_c^2$ is {\em below} the value suggested by the near horizon geometry at extremality. We provide an analytic estimate of $Δ_0$ (numerically, $Δ_0 \approx 0.64$). We also compute analytically the true critical line for the entire range of the scaling dimension. In particular for $q=0$, we obtain an instability down to the unitarity bound. We perform explicit analytic calculations of $T_c$, the condensate and the conductivity. We show that the energy gap in units of $T_c$ diverges as we approach the critical line ($T_c \to 0$).