Small clusters Renormalization Group in 2D and 3D Ising and BEG models with ferro, antiferro and quenched disordered magnetic interactions

The Ising and BEG models critical behavior is analyzed in 2D and 3D by means of a renormalization group scheme on small clusters made of a few lattice cells. Different kinds of cells are proposed for both ordered and disordered model cases. In particular, cells preserving a possible antiferromagnetic ordering under decimation allow for the determination of the N\'eel critical point and its scaling indices. These also provide more reliable estimates of the Curie fixed point than those obtained using cells preserving only the ferromagnetic ordering. In all studied dimensions, the present procedure does not yield the strong disorder critical point corresponding to the transition to the spin-glass phase. This limitation is thoroughly analyzed and motivated.Comment: 14 pages, 12 figure

The Complex Spherical 2+4 Spin Glass: a Model for Nonlinear Optics in Random Media

A disordered mean field model for multimode laser in open and irregular cavities is proposed and discussed within the replica analysis. The model includes the dynamics of the mode intensity and accounts also for the possible presence of a linear coupling between the modes, due, e.g., to the leakages from an open cavity. The complete phase diagram, in terms of disorder strength, source pumping and non-linearity, consists of four different optical regimes: incoherent fluorescence, standard mode locking, random lasing and the novel spontaneous phase locking. A replica symmetry breaking phase transition is predicted at the random lasing threshold. For a high enough strength of non-linearity, a whole region with nonvanishing complexity anticipates the transition, and the light modes in the disordered medium display typical discontinuous glassy behavior, i.e., the photonic glass has a multitude of metastable states that corresponds to different mode-locking processes in random lasers. The lasing regime is still present for very open cavities, though the transition becomes continuous at the lasing threshold.Comment: 26 pages, 13 figure

General phase-diagram of multimodal ordered and disordered lasers in closed and open cavities

We present a unified approach to the theory of multimodal laser cavities including a variable amount of structural disorder. A general mean-field theory is studied for waves in media with variable non-linearity and randomness. Phase diagrams are reported in terms of optical power, degree of disorder and degree of non-linearity, tuning between closed and open cavity scenario's. In the thermodynamic limit of infinitely many modes the theory predicts four distinct regimes: a continuous wave behavior for low power, a standard mode-locking laser regime for high power and weak disorder, a random laser for high pumped power and large disorder and an intermediate regime of phase locking occurring in presence of disorder below the lasing threshold.Comment: 9 pages, 3 figure

Regularization and decimation pseudolikelihood approaches to statistical inference in XYXY-spin models

We implement a pseudolikelyhood approach with l2-regularization as well as the recently introduced pseudolikelihood with decimation procedure to the inverse problem in continuous spin models on arbitrary networks, with arbitrarily disordered couplings. Performances of the approaches are tested against data produced by Monte Carlo numerical simulations and compared also from previously studied fully-connected mean-field-based inference techniques. The results clearly show that the best network reconstruction is obtained through the decimation scheme, that also allows to dwell the inference down to lower temperature regimes. Possible applications to phasor models for light propagation in random media are proposed and discussed.Comment: 10 pages, 12 figure

The glassy random laser: replica symmetry breaking in the intensity fluctuations of emission spectra

The behavior of a newly introduced overlap parameter is analyzed, measuring the correlation between intensity fluctuations of waves in random media in different physical regimes, with varying amount of disorder and non-linearity. Its relationship is established to the standard Parisi overlap order parameter in replica theory for spin-glasses. In the complex spherical spin-glass model, describing the onset and behavior of random lasers, replica symmetry breaking in the intensity fluctuation overlap is shown to occur at high pumping or low temperature. This order parameter identifies the laser transition in random media and describes its glassy nature in terms of emission spectra data, the only data so far accessible in random laser measurements. The theoretical analysis is, eventually, compared to recent intensity fluctuation overlap measurements demonstrating the validity of the theory and providing a straightforward interpretation of different spectral behaviors in different random lasers.Comment: 11 pages, 3 figure

Glassy nature of the hard phase in inference problems

An algorithmically hard phase was described in a range of inference problems: even if the signal can be reconstructed with a small error from an information theoretic point of view, known algorithms fail unless the noise-to-signal ratio is sufficiently small. This hard phase is typically understood as a metastable branch of the dynamical evolution of message passing algorithms. In this work we study the metastable branch for a prototypical inference problem, the low-rank matrix factorization, that presents a hard phase. We show that for noise-to-signal ratios that are below the information theoretic threshold, the posterior measure is composed of an exponential number of metastable glassy states and we compute their entropy, called the complexity. We show that this glassiness extends even slightly below the algorithmic threshold below which the well-known approximate message passing (AMP) algorithm is able to closely reconstruct the signal. Counter-intuitively, we find that the performance of the AMP algorithm is not improved by taking into account the glassy nature of the hard phase. This result provides further evidence that the hard phase in inference problems is algorithmically impenetrable for some deep computational reasons that remain to be uncovered.Comment: 10 pages, 3 figure