Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds

Abstract: Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds $J$ distributed according to a Levy distribution P(J)\propto 1/|J|^{1+\alpha}, |J|>1, are investigated for a range of powers \alpha. The results are compared with those for the Sherrington-Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground state energy densities with \alpha, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (\alpha>2), and compare favorably with recent one-step replica symmetry breaking predictions well below \alpha=2. At and just below \alpha=2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent \omega decays from the SK value \omega_{SK}=2/3 already well above \alpha=2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent \rho that describes the variations of the ground state energy fluctuations with system size decays monotonically from its SK value over the entire range of \alpha and apparently vanishes at \alpha=1.