Numerical results for ground states of spin glasses on Bethe lattices

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) = ESKrootk+1 - (2ESK+root2)/rootk+1, 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 greater than or equal to 9.