Fuzzballs and Random Matrices

TL;DR

This paper investigates whether fuzzball microstates can reproduce the random matrix behavior of black holes, finding that certain profile functions induce level repulsion and spectral features consistent with quantum chaos.

Contribution

It demonstrates that generic fuzzball profiles lead to spectral properties exhibiting level repulsion and linear spectral form factor ramps, bridging fuzzball microstates with black hole chaos.

Findings

Generic profiles induce level repulsion in the spectrum.

Profiles interpolate between Poisson and Wigner-Dyson spectra.

Linear ramp arises from extreme level repulsion in certain limits.

Abstract

Black holes are believed to have the fast scrambling properties of random matrices. If the fuzzball proposal is to be a viable model for quantum black holes, it should reproduce this expectation. This is considered challenging, because it is natural for the modes on a fuzzball microstate to follow Poisson statistics. In a previous paper, we noted a potential loophole here, thanks to the modes depending not just on the $n$-quantum number, but also on the $J$-quantum numbers of the compact dimensions. For a free scalar field $\phi$, by imposing a Dirichlet boundary condition $\phi=0$ at the stretched horizon, we showed that this $J$-dependence leads to a linear ramp in the Spectral Form Factor (SFF). Despite this, the status of level repulsion remained mysterious. In this letter, motivated by the profile functions of BPS fuzzballs, we consider a generic profile $\phi = \phi_0(\theta)$ instead of $\phi=0$ at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients of $\phi_0(\theta)$ are suitably Gaussian distributed), we find that the $J$-dependence of the spectrum exhibits striking evidence of level repulsion, along with the linear ramp. We also find that varying the profile leads to natural interpolations between Poisson and Wigner-Dyson(WD)-like spectra. The linear ramp in our previous work can be understood as arising via an extreme version of level repulsion in such a limiting spectrum. We also explain how the stretched horizon/fuzzball is different in these aspects from simply putting a cut-off in flat space or AdS (ie., without a horizon).