Ground-state structure in a highly disordered spin-glass model

We propose a new Ising spin-glass model on Z^d of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite-volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2^{script N sign}, where script N sign = script N sign(d) is the number of distinct global components in the "invasion forest." We prove that script N sign(d) = ∞ if the invasion connectivity function is square summable. We argue that the critical dimension separating script N sign = 1 and script N sign = ∞ is d_c = 8. When script N sign(d) = ∞, we consider free or periodic boundary conditions on cubes of side length L and show that frustration leads to chaotic L dependence with all pairs of ground states occurring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.