On the number of infinite geodesics and ground states in disordered systems

We study first-passage percolation models and their higher dimensional analogs - models of surfaces with random weights. We prove that under very general conditions the number of lines or, in the second case, hypersurfaces which locally minimize the sum of the random weights is with probability one equal to 0 or with probability one equal to + ∞. As corollaries we show that in any dimension d ≥ 2 the number of ground states of an Ising ferromagnet with random coupling constants equals (with probability one) 2 or + ∞. Proofs employ simple large-deviation estimates and ergodic arguments.