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A statistical mechanic study of the $XY$ model with nonlinear interaction ispresented on bipartite sparse random graphs. The model properties are comparedto those of the $p$-clock model, in which the planar continuous spins arediscretized into $p$ values. We test the goodness of the discrete approximationto the XY spins to be used in numerical computations and simulations and itslimits of convergence in given, $p$-dependent, temperature regimes. The modelsare applied to describe the mode-locking transition of the phases oflight-modes in lasers at the critical lasing threshold. A frequency is assignedto each variable node and function nodes implement a frequency matchingcondition. A non-trivial unmagnetized phase-locking occurs at the phasetransition, where the frequency dependence of the phases turns out to be linearin a broad range of frequencies, as in standard mode-locking multimode laser atthe optical power threshold.

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