Hair on near-extremal Reissner-Nordstr{\o}m AdS black holes

We discuss hairy black hole solutions with scalar hair of mass $m$ and (small) electromagnetic coupling $q^2$, near extremality. Hair forms below a critical temperature $T_c$ and for $q^2 > q_c^2$ where $q_c^2$ is determined by the AdS$_2$ geometry of the horizon and can be negative. At the critical point $q^2 = q_c^2$, the critical temperature vanishes; there is no instability below $q_c^2$. We perform explicit analytic calculations of $T_c$, the condensate and the conductivity for $m^2 =-2$, in which case $q_c^2 = - \frac{1}{4}$. We show that the gap in units of $T_c$ diverges as $T_c \to 0$. We find no discontinuity in the behavior of the system across $q^2 = 0$.