  Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have
exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to
terms that vanish as $n\to\infty$. We show that {\it two independent biased
random walks} executed by the polar angle determine the trajectory of the cell
perimeter. We find the limit distribution of (i) the angle between two
successive vertex vectors, and (ii) the one between two successive perimeter
segments. We obtain the probability law for the perimeter's long wavelength
deviations from circularity. We prove Lewis' law and show that it has
coefficient 1/4.
