  The quantum-limited linewidth of a laser cavity is enhanced above the
Schawlow-Townes value by the Petermann factor K, due to the non-orthogonality
of the cavity modes. We derive the relation between the Petermann factor and
the residues of poles of the scattering matrix and investigate the statistical
properties of the Petermann factor for cavities in which the radiation is
scattered chaotically. For a single scattering channel we determine the
complete probability distribution of K and find that the average Petermann
factor $<K>$ depends non-analytically on the area of the opening, and greatly
exceeds the most probable value. For an arbitrary number N of scattering
channels we calculate $<K>$ as a function of the decay rate $\Gamma$ of the
lasing mode. We find for $N\gg 1$ that for typical values of $\Gamma$ the
average Petermann factor $<K>\propto \sqrt{N}\gg 1$ is parametrically larger
than unity.
