Three-dimensional spin glasses resist clean analytical solutions, and numerical methods that probe their critical behavior typically require comparing many replicas — an expensive proposition. The spectral approach constructs overlap matrices from two-dimensional cross-sections and examines their eigenvalue distributions. At high temperatures, the distribution follows the Wigner semicircle law — the universal signature of uncorrelated random matrices. As temperature drops toward the spin-glass critical point, the distribution deforms continuously into a Gaussian, passing through intermediate shapes described by Tsallis q-statistics with the entropic index q sliding from -1 to 1.
The transition in spectral shape is not just a diagnostic convenience. It reveals that the paramagnetic phase — often treated as structurally featureless — already contains statistical signatures of the approaching transition. The spectral density carries information about incipient correlations that bulk thermodynamic quantities miss. Local level statistics remain GOE-universal throughout; temperature reshapes only the global envelope, leaving the fine structure untouched.
The principle: critical transitions can be read from the shape of a spectrum before they manifest in the order parameter. The global distribution of eigenvalues functions as a more sensitive thermometer than the quantities traditionally used to locate phase boundaries — detecting structure in what appears, by other measures, to be noise.
(arXiv:2603.03513)