  We consider an ensemble of $n$ nonintersecting Brownian particles on the unit
circle with diffusion parameter $n^{-1/2}$, which are conditioned to begin at
the same point and to return to that point after time $T$, but otherwise not to
intersect. There is a critical value of $T$ which separates the subcritical
case, in which it is vanishingly unlikely that the particles wrap around the
circle, and the supercritical case, in which particles may wrap around the
circle. In this paper, we show that in the subcritical and critical cases the
probability that the total winding number is zero is almost surely 1 as
$n\to\infty$, and in the supercritical case that the distribution of the total
winding number converges to the discrete normal distribution. We also give a
streamlined approach to identifying the Pearcey and tacnode processes in
scaling limits. The formula of the tacnode correlation kernel is new and
involves a solution to a Lax system for the Painlev\'{e} II equation of size 2
$\times$ 2. The proofs are based on the determinantal structure of the
ensemble, asymptotic results for the related system of discrete Gaussian
orthogonal polynomials, and a formulation of the correlation kernel in terms of
a double contour integral.
