Numerical results for ground states of spin glasses on Bethe lattices

摘要

The average ground state energy and entropy for ±J spin glasses on Bethe lattices of connectivities k+1 = 3…, 26 at T = 0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n = 2~(12)), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_(k+1) but also for their entropies s_(k+1). The results indicate sizable differences between lattices of even and odd connectivities. The extrapolated ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k+1 to within a fraction of a percent onto a simple functional form, e_(k+1) = E_(SK)(k+1)~(1/2) - (2E_(SK) + 2~(1/2))/(k+1)~(1/2), where E_(SK) = -0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k+1, which do not distinguish between even and odd connectivities. We also find non-zero entropies per spin s_(k+1) at small connectivities. While s_(k+1) seems to vanish asymptotically with 1/(k+1) for even connectivities, it is numerically indistinguishable from zero already for odd k+1 ≥ 9.