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

Abstract

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 the planar continuous spins are discretized into $p$ values. We test the goodness of the discrete approximation to the XY spins to be 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 non-trivial unmagnetized phase-locking occurs at the phase transition, where the frequency dependence of the phases turns out to be linear in a broad range of frequencies, as in standard mode-locking multimode laser at the optical power threshold.

13 pages, 13 figures

References in corpus (4)

• General phase-diagram of multimodal ordered and disordered lasers in closed and open cavities
• Phase diagram and complexity of mode-locked lasers: from order to disorder
• Mode-locking transitions in nano-structured weakly disordered lasers
• Phase transitions and topology in 2+k XY mean-field models

Cited by in corpus (5)

• The Complex Spherical 2+4 Spin Glass: a Model for Nonlinear Optics in Random Media
• Approximating the XY model on a random graph with a $q$-state clock model
• Statistical physics of nonlinear wave interaction
• Statistical physical theory of mode-locking laser generation with a frequency comb
• Inference for interacting linear waves in ordered and random media