Extremal statistics in the energetics of domain walls

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 ΔE1∼Lθf(Nz), where f(y)∼[lny]−1/2, θ is the energy fluctuation exponent, L is the length scale, and Nz 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.