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

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Cited by in corpus (15)

• The Complex Spherical 2+4 Spin Glass: a Model for Nonlinear Optics in Random Media
• Statistical mechanics models for multimode lasers and random lasers
• Approximating the XY model on a random graph with a $q$-state clock model
• Statistical physics of nonlinear wave interaction
• Self-starting nonlinear mode-locking in random lasers
• Statistical physical theory of mode-locking laser generation with a frequency comb
• The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism
• Comparison of Gabay-Toulouse and de Almeida-Thouless instabilities for the spin glass XY model in a field on sparse random graphs
• Regularization and decimation pseudolikelihood approaches to statistical inference in $XY$-spin models
• Multi-body quenched disordered $XY$ and $p$-clock models on random graphs
• Low energy excitations of mean-field glasses
• Improved Pseudolikelihood Regularization and Decimation methods on Non-linearly Interacting Systems with Continuous Variables
• Inverse problem for multi-body interaction of nonlinear waves
• Inference for interacting linear waves in ordered and random media
• Mean-field theory of vector spin models on networks with arbitrary degree distributions