The emergence of the

Context. There is still much uncertainty around the mechanism that rules the accretion of proto-planetary disks. In recent years, magnetohydrondynamic (MHD) wind-driven accretion has been proposed as a valid alternative to the more conventional viscous accretion. In particular, winds have been shown to reproduce the observed correlation between the mass of the disk MD and the mass accretion rate onto the central star Ṁ*, but this has been done only for specific conditions. It is not clear whether this implies fine tuning or if it is a general result. Aims. We investigated under which conditions the observed correlation between the mass of the disk MD and the mass accretion rate onto the central star Ṁ*, can be obtained. Methods. We present mainly analytical calculations, supported by Monte Carlo simulations. We also perform a comparison with the observed data to test our predictions. Results. In the absence of a correlation between the initial mass M0 and the initial accretion timescale tacc,0 , we find that the slope of the MD − Ṁ* correlation depends on the value of the spread of the initial conditions of masses and lifetimes of disks. Then we clarify the conditions under which a disk population can be fitted with a single power law. Moreover, we derive an analytical expression for the spread of log(MD/Ṁ*) valid when the spread of tacc is taken to be constant. In the presence of a correlation between M0 and tacc,0, we derive an analytical expression for the slope of the MD−Ṁ* correlation in the initial conditions of disks and at late times. In this new scenario, we clarify under which conditions the disk population can be fitted by a single power law, and we provide empirical constraints on the parameters ruling the evolution of disks in our models. Conclusions. We conclude that MHD winds can predict the observed values of the slope and the spread of the MD−Ṁ* correlation under a broad range of initial conditions. This is a fundamental expansion of previous works on the MHD paradigm, exploring the establishment of this fundamental correlation beyond specific initial conditions.