Extremal Optimization for 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) / 1/|J|1+α; |J| > 1; are investigated for a range of powers α. We determine ground state energy density variation with α and their finite-size corrections. We find that the energies attain universally the Parisi-energy of the SK as long as the second moment of P (J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ωSK = = 2/3 already well above α = 2, at which point it reaches a minimum.