Statistical physics of nonlinear wave interaction

The thermodynamic properties of vector [O(2) and complex spherical] models with four-body interactions are analyzed. When defined in dense topologies, these are effective models for the nonlinear interaction of scalar fields in the presence of a stochastic noise, as has been well established for the case of the mode-locking laser formation in a closed cavity. With the help of an efficient Monte Carlo algorithm we show how, beyond the fully connected case, rich phenomenology emerges. Below a certain dilution threshold, the spherical model condenses in a nonequipartite way, while in the XY model the transition becomes continuous and the O(2) symmetry remains unbroken. We attribute this fact to the invariance under gauge transformations. The introduction of topological inhomogeneities in the network of quadruplets induces some features: again symmetry conservation; the vanishing of two-point correlators; and a dynamical correlation function presenting two time scales, the large one being related to the transition between different degenerated configurations, connected by particular gauge transformations. We discuss possible experimental implications of these results in the context of nonlinear optics.