(Alice K. Harding, Demosthenes Kazanas)
A decade of Fermi operation has provided a plethora of observational data that shifted the study of gamma-ray pulsars from discovery to astronomy. Our studies of macroscopic pulsar magnetosphere models, guided by the Fermi observations, have revealed not only the equatorial current sheet beyond the light-cylinder as the main dissipative region where the observed gamma-ray emission is produced (FIDO models) but also a relation between the local plasma conductivity and the spin-down power. Our global PIC models confirmed the equatorial current sheet as the site of the gamma-ray emission especially for the high particle injection rates and high spin-down powers and uncovered a relation between the global particle injection rate (F) and the spin-down power. Nonetheless, in these models, the originally assumed uniform (along the different magnetic field lines) particle injection provided, for the low F values, model γ-ray light-curves that are not always consistent with the observed ones. In the second generation of PIC models, we applied magnetic field line dependent particle injection and found that the particle population that regulates the observed gamma-ray emission is the one that is injected near the separatrix that separates the open from the closed field lines. By controlling the geometric features of this region and the corresponding particle injection, we are able to reproduce (not only broadly but also more accurately than ever before) the observed Fermi gamma-ray phenomenology of the millisecond and young pulsars for the entire range of spin-down powers. I will discuss the implication of these results, which deepen considerably our understanding of the physical mechanisms behind the high-energy emission in pulsar magnetospheres.