  We study at T=0 the minimum energy of a domain wall and its gap to the first
excited state concentrating on two-dimensional random-bond Ising magnets. The
average gap scales as $\Delta E_1 \sim L^\theta f(N_z)$, where $f(y) \sim [\ln
y]^{-1/2}$, $\theta$ is the energy fluctuation exponent, $L$ length scale, and
$N_z$ the number of energy valleys. The logarithmic scaling is due to extremal
statistics, which is illustrated by mapping the problem into the
Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of
domain walls has also a logarithmic dependence on system size.
