Ground states of the Sherrington-Kirkpatrick spin glass with Levy bonds

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) proportional to 1/vertical bar J vertical bar(1+alpha), vertical bar J vertical bar > 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 > 42). They 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 putative 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 for decreasing alpha and appears to vanish at alpha = 1.