Volcano transition in populations of phase oscillators with random nonreciprocal interactions

Populations of heterogeneous phase oscillators with frustrated random interactions exhibit a quasi-glassy state in which the distribution of local fields is volcano-shaped. In a recent work [Phys. Rev. Lett. 120, 264102 (2018)] the volcano transition was replicated in a solvable model using a low-rank, random coupling matrix $\mathbf{M}$. We extend here that model including tunable nonreciprocal interactions, i.e. ${\mathbf{M}}^T\mathrm{\ne }\mathbf{M}$. 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.