Hair on near-extremal Reissner-Nordstrom AdS black holes

We discuss hairy black hole solutions with scalar hair of scaling dimension Delta 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 (Delta, q(2)) above a critical line q(c)(2)(Delta). For Delta > Delta(0), the critical coupling q(c)(2) is determined by the AdS(2) geometry of the horizon. For Delta < Delta(0), q(c)(2) is below the value suggested by the near horizon geometry at extremality. We provide an analytic estimate of Delta(0) (numerically, Delta(0) approximate to 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 -> 0).