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A bstract 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 ϕ , by imposing a Dirichlet boundary condition ϕ = 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 ϕ = ϕ 0 ( θ ) instead of ϕ = 0 at the stretched horizon. For various notions of genericity (eg. when the Fourier coefficients of ϕ 0 ( θ ) 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 (i.e., without a horizon).
Das et al. (Thu,) studied this question.