The Anderson model places a quantum particle on a graph with random on-site potentials. When the disorder is weak, the particle delocalizes — its wavefunction spreads across the graph. When the disorder is strong, the particle localizes — the wavefunction decays exponentially from a single site. The transition between these regimes, the mobility edge, is one of the fundamental objects in condensed matter physics.
On regular lattices (Z^d for d ≥ 3), the mobility edge separates the spectrum into a localized band edge and a delocalized center. On the Bethe lattice — an infinite tree with fixed branching — the spectrum is fully delocalized at weak disorder and fully localized at strong disorder. Random regular graphs sit between these limits, sharing local structure with the Bethe lattice but having finite size and cycles.
Liu and Lopatto establish the phase diagram: for sufficiently large degree and Gaussian disorder, the spectrum has a finite delocalized interval sandwiched between two unbounded localized regions. The conductor lives in the middle. The insulator flanks it on both sides.
The proof leverages the Bethe lattice as the local limit of the random regular graph. Results from Aggarwal-Lopatto (2025) characterize the Bethe lattice spectrum, and the new work extends these to finite graphs through preliminary estimates that control the error introduced by cycles and finite size.
The sandwich structure is the key geometric insight. Unlike Z^d, where the mobility edge separates one localized region from one delocalized region, the random regular graph has localization on both sides — at both spectral extremes. The delocalized phase is bounded. There is no escape to infinity in either direction. The particle can propagate only within a finite spectral window, trapped between two insulators.