Volcano transition in populations of phase oscillators with random nonreciprocal interactions

Populations of heterogeneous phase oscillators with frustrated random interactions exhibit a quasiglassy state in which the distribution of local fields is volcanoshaped. In a recent work [Phys. Rev. Lett. 120, 264102 (2018)10.1103/PhysRevLett.120.264102], the volcano transition was replicated in a solvable model using a low-rank, random coupling matrix M. We extend here that model including tunable nonreciprocal interactions, i.e., M^{T}≠M. More specifically, we formulate two different solvable models. In both of them the volcano transition persists if matrix elements M_{jk} and M_{kj} are enough correlated. Our numerical simulations fully confirm the analytical results. To put our work in a wider context, we also investigate numerically the volcano transition in the analogous model with a full-rank random coupling matrix.