  The Sinai model of a tracer diffusing in a quenched Brownian potential is a
much studied problem exhibiting a logarithmically slow anomalous diffusion due
to the growth of energy barriers with the system size. However, if the
potential is random but periodic, the regime of anomalous diffusion crosses
over to one of normal diffusion once a tracer has diffused over a few periods
of the system. Here we consider a system in which the potential is given by a
Brownian Bridge on a finite interval $(0,L)$ and then periodically repeated
over the whole real line, and study the power spectrum $S(f)$ of the diffusive
process $x(t)$ in such a potential. We show that for most of realizations of
$x(t)$ in a given realization of the potential, the low-frequency behavior is
$S(f) \sim {\cal A}/f^2$, i.e., the same as for standard Brownian motion, and
the amplitude ${\cal A}$ is a disorder-dependent random variable with a finite
support. Focusing on the statistical properties of this random variable, we
determine the moments of ${\cal A}$ of arbitrary, negative or positive order
$k$, and demonstrate that they exhibit a multi-fractal dependence on $k$, and a
rather unusual dependence on the temperature and on the periodicity $L$, which
are supported by atypical realizations of the periodic disorder. We finally
show that the distribution of ${\cal A}$ has a log-normal left tail, and
exhibits an essential singularity close to the right edge of the support, which
is related to the Lifshitz singularity. Our findings are based both on analytic
results and on extensive numerical simulations of the process $x(t)$.
