  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).
