Scaling laws and relaxation rate at desynchronization in coupled oscillators

Complex interactions leading to phase transitions continue to hold a due interest in the scientific community. We charactersize a phase transition in a coupled oscillators model where interactions are not local in nature. At a first order transition i.e an abrupt jump in order parameter, we obtain the order parameter scaling exponent $1/2$ and show that the relaxation time diverges near the transition point, both numerically and analytically. We also verify a finite time finite size scaling by considering a correlation measure in the discrete set of all interacting units. Due to divergence in correlation measure at the transition point, the order parameter becomes a homogeneous function of all the relevent parameters. The homogeneity assumption is numerically verified using data collapse method. Various finite size scaling relations and exponents are also obtained. This study suggests that presence of non local interactions may allow for critical behaviour at a first order transition.