  Up to now there has been no reliable method to calculate the Casimir force
when surface roughness becomes comparable with the separation between bodies.
Statistical analysis of rough Au films demonstrates rare peaks with heights
considerably larger than the root-mean-square (rms) roughness. These peaks
define the minimal distance between rough surfaces and can be described with
extreme value statistics. We show that the contributions of high peaks to the
force can be calculated independently of each other while the contribution of
normal roughness can be evaluated perturbatively beyond the proximity force
approximation. The developed method allows a reliable force estimation for
short separations. Our model explains the strong hitherto unexplained deviation
from the normal Casimir scaling observed experimentally at short separations.
