The Spectral Edge at Large Central Charge and Black Hole Statistics

We introduce a new approach to high-energy spectra in AdS${}_3$/CFT${}_2$, addressing the open challenge of incorporating spectral statistics into the modular bootstrap. Using the analytic theory of $L$-functions attached to two-dimensional CFTs, we leverage modular constraints among high-energy states to derive a sum rule relating the spectral edge to bulk spectral fluctuations. As an application, we establish a refined modular bootstrap bound: letting ${\mathrm{\Delta }}_1$ be the conformal dimension of the lightest excited Virasoro primary of a unitary CFT of central charge $c$, we show that ${\mathrm{\Delta }}_1-\frac{c-1}{12}\le o(c)$, with at most one exceptional value of the scalar gap at each sufficiently large $c$; moreover, along any sequence in $c$ for which such exceptional gaps are realized, the remaining gaps obey the stronger bound ${\mathrm{\Delta }}_1-\frac{c-1}{12}\le O(1)$. The result applies to theories with logarithmic rigidity of spectral counting fluctuations, as in random matrix theory.