Voronoi–graph spectrum of disordered phases

Disordered phases, from liquids and glasses to jammed packings and living cell layers, lack long-range order yet differ structurally in ways that standard two-point measures may miss. Here, we characterize such states using the spectrum of the Voronoi–Delaunay neighbor graph. Specifically, we study the density of eigenvalues for the graph Laplacian and the moments of its adjacency matrix. This dimension-agnostic, reciprocal-space fingerprint, whose construction involves no adjustable parameters, splits into low-frequency descriptors (gap, spectral dimension), set by large-scale connectivity, and moments (coordination variance, triangle counts), set by the local environment. The two are shown to be complementary. We apply our methods to simulated models in two and three spatial dimensions and to experiments with living monolayers of swarming bacteria. Simulations for the random-organization transition show that the low-frequency descriptors are blind to the hyperuniform order, while the moments successfully pinpoint the critical point. Analyzing experiments of monolayer bacterial swarms, the spectrum finds a structural transition for elongated mutants as a function of area fraction, but not for the wild-type cells. This is in accordance with previous results obtained from temporal fluctuations. The spectrum thus reads structure, classifies disordered phases, and pinpoints structural phase transitions.