Extremal statistics in the energetics of domain walls

We study at $T=0\mathrm{}$ 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 $f\left(y\right)\mathrm{}\sim [\ln y{]}^{-1/2},$ θ is the energy fluctuation exponent, L is the length scale, and $N_z$ is 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 also has a logarithmic dependence on the system size.