  Lasing transition is known to be analogous to the second-order phase
transition. Furthermore, for some cases, it is possible to define the
Ginzburg-Landau (GL) potential, and the GL theory predicts the photon
statistical properties of lasers. However, the GL potential for lasers is
surprising, because lasers are operating far from equilibrium. In this paper,
we theoretically examine the validity of the GL theory for lasers in terms of
various parameters, particularly, the ratio between photon and carrier
lifetimes. For this purpose, we use stochastic rate equations and higher-order
photon correlation functions. With higher-order photon correlation
measurements, we can check whether or not laser dynamics are described by the
GL theory. We demonstrate that, for low-$\beta$ lasers, the GL theory is
applicable even when the photon lifetime is comparable to the carrier lifetime
and that photon-carrier relaxation oscillation is the fundamental origin of the
breakdown of the GL theory, which can be understood in the framework of center
manifold reduction.
