Before revision:  When the arrangement is obtained from any fixed projective line arrangement of n lines by choosing a random cellto contain the south-pole, we prove an upper bound of O((k+1)^2) on the expected complexity of the k-level .
After revision:  When the arrangement is obtained from any fixed projective line arrangement of n lines by choosing a random cellto contain the south-pole, wesouth pole prove an upper bound of O((k+1)^2) on the expected complexity of the k-level .