= Small-radius Brownian winding law
{title2=$\theta_\varepsilon/\log\varepsilon\Rightarrow C/(2\pi)$}
= Spitzer small-radius winding law
{c}
{synonym}
For planar <Brownian motion> started at radius $\varepsilon$ and stopped on first hitting radius one, the net winding count divided by $\log\varepsilon$ converges to a standard <Cauchy random variable> divided by $2\pi$. The complex exponential maps Brownian coordinates in the left half-plane to the punctured unit disk after the <conformal Brownian clock>. The unwrapped argument at exit is the vertical half-plane exit coordinate, with the <Cauchy exit law from a Brownian half-plane>. Counting integer turns introduces only a uniformly bounded error in winding units.
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