A random-matrix theory is presented for the linewidth of a laser cavity in
which the radiation is scattered chaotically. The linewidth is enhanced abo
ve the Schawlow-Townes value by the Petermann factor K, due to the nonortho
gonality of the cavity modes. The factor K is expressed in terms of a non-H
ermitian random matrix, and its distribution is calculated exactly for the
case in which the cavity is coupled to the outside via a small opening. The
average of K is found to depend nonanalytically on the area of the opening
, and to greatly exceed the most probable value.