God of the gaps: random matrix models and the black hole spectral gap

A bstract We show that random matrix models are a natural tool for understanding the appearance of a large gap in the microstate spectrum of black holes when there is a high degeneracy of states, in a variety of settings. While the most natural context is extended supersymmetry, where the number of BPS states scales as $$ {\textrm{e}}^{S_0} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>e</mml:mi> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:msup> </mml:math> , where S 0 is the T = 0 entropy, it is a robust feature that a large gap will appear whenever there is a mechanism producing a high degree of degeneracy. In random matrix model terms, the phenomenon is simply an extreme case of eigenvalue repulsion in the effective log gas description. We exhibit several examples for illustration, starting with the simple Wishart model, continuing with extensions of it that incorporate multicritical behaviour allowing for the emergence of gravity, and culminating in constructing multicritical matrix models of $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> and $$ \mathcal{N}=4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:math> JT supergravity theories, the latter of which is new.