Energy exponents and corrections to scaling in Ising spin glasses

We study the probability distribution $P(E)$ of the ground-state energy E in various Ising spin glasses. In most models, $P(E)$ seems to become Gaussian with a variance growing as the system's volume V. Exceptions include the Sherrington-Kirkpatrick model (where the variance grows more slowly, perhaps as the square root of the volume), and mean-field diluted spin glasses having $\ifmmode\pm\else\textpm\fi{}J$ couplings. We also find that the corrections to the extensive part of the disorder averaged energy grow as a power of the system size; for finite-dimensional lattices, this exponent is equal, within numerical precision, to the domain-wall exponent ${\ensuremath{\theta}}_{\mathrm{DW}}.$ We also show how a systematic expansion of ${\ensuremath{\theta}}_{\mathrm{DW}}$ in powers of ${e}^{\ensuremath{-}d}$ can be obtained for Migdal-Kadanoff lattices. Some physical arguments are given to rationalize our findings.