We find a significant destructive interference among Kerr overtones in the
early ringdown induced by an extreme mass-ratio merger of a massive black hole
and a compact object, and that the ringdown spectrum apparently follows the
Fermi-Dirac distribution. We numerically compute the spectral amplitude of
gravitational waves induced by a particle plunging into a Kerr black hole and
study the excitation of multiple quasi-normal (QN) modes. We find that the
start time of ringdown is before the strain peak of the signal and corresponds
to the time when the particle passes the photon sphere. When the black hole has
the near-extremal rotation, the Kerr QN frequencies are close to the fermionic
Matsubara frequencies with the Hawking temperature and the chemical potential
of the superradiant frequency. We indeed find that the absolute square of the
spectral amplitude apparently follows the Fermi-Dirac distribution with the
chemical potential of around the real QN frequency of the fundamental mode.
Fitting the Boltzmann distribution to the data in higher frequencies, the
best-fit temperature is found out to be close to the Hawking temperature,
especially for rapid rotations. In the near-extremal limit, the
gravitational-wave spectrum exhibits a would-be Fermi degeneracy with the Fermi
surface at the superradiant frequency $\omega = \mu_{\rm H}$. We show that the
greybody factor, i.e., the absorption cross section of a black hole, leads to
the Fermi-Dirac distribution. As the greybody factor is another no-hair
quantity of black holes, this opens a new possibility that we can test general
relativity by observationally searching for the Boltzmann distribution in
$\omega \gtrsim \mu_{\rm H}$ without extracting QN modes from ringdown. We
could measure the mass and angular momentum of ringing black holes and could
probe the Kerr/CFT by measuring the greybody factor imprinted on the ringdown
spectrum.
