  In a recent work the present authors have shown that the eigenvalue
probability density function for Dyson Brownian motion from the identity on
$U(N)$ is an example of a newly identified class of random unitary matrices
called cyclic P\'olya ensembles. In general the latter exhibit a structured
form of the correlation kernel. Specialising to the case of Dyson Brownian
motion from the identity on $U(N)$ allows the moments of the spectral density,
and the spectral form factor $S_N(k;t)$, to be evaluated explicitly in terms of
a certain hypergeometric polynomial. Upon transformation, this can be
identified in terms of a Jacobi polynomial with parameters $(N(\mu - 1),1)$,
where $\mu = k/N$ and $k$ is the integer labelling the Fourier coefficients.
From existing results in the literature for the asymptotics of the latter, the
asymptotic forms of the moments of the spectral density can be specified, as
can $\lim_{N \to \infty} {1 \over N} S_N(k;t) |_{\mu = k/N}$. These in turn
allow us to give a quantitative description of the large $N$ behaviour of the
average $ \langle | \sum_{l=1}^N e^{ i k x_l} |^2 \rangle$. The latter exhibits
a dip-ramp-plateau effect, which is attracting recent interest from the
viewpoints of many body quantum chaos, and the scrambling of information in
black holes.
