Nonlinear XY and p-clock models on sparse random graphs: Mode-locking transition of localized waves

A statistical mechanic study of the XY model with nonlinear interaction is presented on bipartite sparse random graphs. The model properties are compared to those of the p-clock model, in which planar continuous spins are discretized into p values. We test the goodness of the discrete approximation to XY spins used in numerical computations and simulations and its limits of convergence in given, p-dependent temperature regimes. The models are applied to describe the mode-locking transition of the phases of light modes in lasers at the critical lasing threshold. A frequency is assigned to each variable node, and function nodes implement a frequency matching condition. A nontrivial unmagnetized phase-locking occurs at the phase transition, where the frequency dependence of the phases turns out to be linear over a broad range of frequencies, as in a standard mode-locking multimode laser at the optical power threshold.