  We detail the derivation of nanolaser models that include coherent and
incoherent variables and predict the existence of a laser threshold,
irrespective of cavity size and emitter number, for both single- and
multi-electron systems. The growth in photon number in the lasing mode is
driven by an increase in correlation between absorption and emission processes,
leading to the onset of self-sustained stimulated emission (laser threshold),
followed, in turn, by a correlation decrease and ending with the dominance of
coherent emission. The first-order coherence $g^{(1)}$ steadily increases, as
the pump grows towards the laser threshold value, and reaches unity at or
beyond threshold. The transition toward coherent emission becomes increasingly
sharp as the number of emitters and of the coupled electromagnetic cavity modes
increase, continuously connecting, in the thermodynamic limit, the physics of
nano- and macroscopic lasers at threshold. Our predictions are in remarkable
agreement with experiments whose first-order coherence measurements have so far
been explained only phenomenologically. A consistent evaluation of different
threshold indicators provides a tool for a correct interpretation of
experimental measurements at the onset of laser action.
