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.

• Received 21 November 2014
• Revised 30 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.054201