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 $2^{\cal N}$, where ${\cal N} = {\cal N}(d)$ is the number of distinct global components in the ``invasion forest''. We prove that ${\cal N}(d) = \infty$ if the invasion connectivity function is square summable. We argue that the critical dimension separating ${\cal N} = 1$ and ${\cal N} = \infty$ is $d_c = 8$. When ${\cal N}(d) = \infty$, we consider free or periodic boundary conditions on cubes of side length $L$ and show that frustration leads to chaotic $L$ dependence with {\it all} pairs of ground states occuring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.