Random translation-invariant Hamiltonians and their spectral gaps

We consider random translation-invariant frustration-free quantum spin Hamiltonians on $\mathbb Z^D$ in which the nearest-neighbor interaction every direction is randomly sampled and then distributed across lattice. Our main result that, under a small rank constraint, are automatically they gapped with positive probability. This extends previous results 1D chains to all dimensions. The argument additionally controls local gap. As an application, we obtain 2D area law for cut-dependent ground state via recent AGSP methods of Anshu-Arad-Gosset.