This paper studies how adding a small random potential changes the spectrum of a quantum-like model on a hyperbolic lattice made from squares. The underlying graph is the Cartesian product of the usual integer lattice Z^d with a regular hyperbolic square tiling where five squares meet at each vertex (instead of four, as on the flat square lattice). The authors show that, at weak disorder, the model has extended spectral states: locally the spectrum has a positive density that is absolutely continuous and there is no singular part on a set of energies of positive measure.

The model is the standard Anderson operator: the graph adjacency operator plus a random potential that is independent and identically distributed at each vertex, scaled by a disorder strength λ. Under mild assumptions on the single-site probability density (bounded support and some lower bound on an interval containing 0), they prove there is a deterministic set of energies D of positive Lebesgue measure such that, almost surely for a given random realization, every local spectral measure has a positive absolutely continuous density on almost every energy in D and no singular component. Concretely, one of the main statements covers small λ (for the pure hyperbolic square lattice without a Z factor the paper gives λ up to about 10^-4, while for any positive Z^d factor it gives λ up to 1/5). If the single-site density is bounded below across the whole support [-1,1], the allowed disorder range is larger and includes the common case λ≤2 cited in the paper.

The authors also obtain stronger interval results for specific distributions. For truncated Cauchy site distributions they prove purely absolutely continuous spectrum on whole energy intervals (with positive local density almost everywhere). These interval conclusions are stable: they remain true if the Cauchy law is replaced by a compactly supported density that is sufficiently close in total variation and still bounded below on an appropriate interval. The paper notes that these intervals lie outside the spectrum of the free (non-random) adjacency operator.