Extremal Scalarization of Charged Black Holes: Miransky Scaling across Reissner-Nordström Extremality

Abstract: We study spontaneous scalarization near and at extremality in Einstein-Maxwell-scalar theory. As Reissner-Nordström (RN) extremality is approached from below, the nonextremal scalarization existence curves collapse toward a common critical threshold $α_c=1/4$ given by the Breitenlohner-Freedman bound for the near-horizon throat (AdS$_2\times S<sup>2$) of its extremal black hole. For a polynomial coupling, we construct extremal black holes with nonconstant scalar hair, satisfying $M<sup>2+Q_s<sup>2=Q<sup>2$, which places them in the overcharged region $Q/M&gt;1$ beyond the RN bound. A common near-region phase mechanism predicts accumulation toward the RN extremality from both sides, yielding Miransky-type scaling. In the extremal sector, the logarithmic phase interval doubles, halving the exponent which denotes the geometric spacing of successive node scales.