Ground States of the Mean-Field Spin Glass with 3-Spin Couplings

We use heuristic optimization methods in extensive computations to determine with low systematic error ground-state configurations of the mean-field p-spin glass model with p=3. Here, all possible triplets in a system of N Ising spins are connected with a bond. This model has been of recent interest, since it exhibits the "overlap gap condition," which should make it prohibitive to find ground states asymptotically with local search methods when compared, for instance, with the p=2 case better known as the Sherrington-Kirkpatrick model (SKM). Indeed, it proves more costly to find good approximations for p=3 than for the SKM, even for our heuristic. Compared to the SKM, the ground-state behavior for p=3 is quite distinct also in other ways. For the SKM, finite-size corrections for large system sizes N→∞, of both the ensemble average over ground-state energy densities and the width of their distribution, vary anomalously with noninteger exponents. In the p=3 case here, the energy density and its distribution appear to scale with lnN/N and 1/N corrections, respectively. The distribution itself is consistent with a Gumbel form. Even more stark is the contrast for the bond-diluted case, where the SKM has shown previously a notable variation of the anomalous corrections exponent with the bond density, while for p=3 no such variation is found here. Hence, for the 3-spin model, all measured corrections scale the same as for the random energy model (REM), corresponding to p=∞. This would suggest that all p-spin models with p≥3 exhibit the same ground-state corrections as in the REM.