Spectral properties and phase diagrams of sparse antagonistic random matrices with diagonal disorder and Jacobian-like structure

Abstract

Complex interacting systems are often modelled by random matrices whose spectral properties dictate stability. In sparse antagonistic matrices without diagonal disorder, low connectivity gives rise to a characteristic reentrance effect in the spectral boundary near the real axis, which disappears via a continuous transition as the connectivity increases. The reentrance effect implies the presence of a complex leading eigenvalue, which suggests the existence of a phase characterized by oscillatory dynamics around equilibrium. Here, we expand the investigation to matrices featuring diagonal disorder and a Jacobian-like structure. In these settings, the spectrum also develops a segment of eigenvalues accumulating on the real axis, which can trigger a discontinuous jump of the complex leading eigenvalue to a purely real value. The interplay between connectivity and disorder produces a rich variety of spectral behaviours. Employing the cavity method and a an adaptation of the Population Dynamics algorithm, we map a phase diagram with five distinct spectral phases. Finally, we show that the algorithm underestimates the spectral support under strong disorder, motivating future technical developments to handle this limit.