The statistics of the ground-state and domain-wall energies for the
two-dimensional random-bond Ising model on square lattices with independent,
identically distributed bonds of probability p of Jijβ=β1 and (1βp) of
Jijβ=+1 are studied. We are able to consider large samples of up to
3202 spins by using sophisticated matching algorithms. We study LΓL
systems, but we also consider LΓM samples, for different aspect ratios
R=L/M. We find that the scaling behavior of the ground-state energy and
its sample-to-sample fluctuations inside the spin-glass region (pcββ€pβ€1βpcβ) are characterized by simple scaling functions. In particular, the
fluctuations exhibit a cusp-like singularity at pcβ. Inside the spin-glass
region the average domain-wall energy converges to a finite nonzero value as
the sample size becomes infinite, holding R fixed. Here, large finite-size
effects are visible, which can be explained for all p by a single exponent
Οβ2/3, provided higher-order corrections to scaling are included.
Finally, we confirm the validity of aspect-ratio scaling for Rβ0: the
distribution of the domain-wall energies converges to a Gaussian for Rβ0,
although the domain walls of neighboring subsystems of size LΓL are
not independent.Comment: 11 pages with 15 figures, extensively revise