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Evidence of an emergent pseudo-localized phase characterizing the low-temperature replica symmetry breaking phase of the complex disordered models for glassy light is provided in the mode-locked random laser model. A pseudo-localized phase corresponds to a state in which the intensity of light modes is neither equipartited among all modes nor really localized on few of them. Such a hybrid phase has been recently characterized in other models, such as the Discrete Non-Linear Schr\"odinger equation, just as a finite size effect, while in the low temperature phase of the glassy random laser it seems to be robust in the limit of large size., Comment: 21 pages, 6 figures

The measurement of the optical Transmission Matrix (TM) enables to access "open channels": input patterns, specific to each scattering structure, capable to deliver very high transmission. Various approaches, based either on multiple interferometric measurements or on systematic random testing of incident wavefronts, enable to estimate the inputs required to excite these open channels. Here, we provide for the first time an approach enabling the complete and reference-less retrieval of the open channels. It is based on the full mapping all the pairwise interference terms resulting from all the input modes couples. We show that these interference terms are organized into a bi-dyadic coupling matrix whose eigenvalues enables to access the open channel. A disordered optical system, is thus behaving exactly like an Hopfield neural network, where a specific input vector (an eigenvalue of the neurons' coupling matrix) enables to retrieve a specific memory pattern. The proposed Hopfield like open-channel-retrieval approach, enables to reach almost 100$\%$ of the theoretically expected value of the Intensity. Moreover employing a digital micromirror device to modulate light, we demonstrate high speed laser scanning at the back of a disordered medium., 6 pages, 5 figures

By means of enhanced Monte Carlo numerical simulations parallelized on GPU's we study the critical properties of the spin-glass-like model for the mode-locked glassy random laser, a $4$-spin model with complex spins with a global spherical constraint and quenched random interactions. Using two different boundary conditions for the mode frequencies we identify the critical points and the critical indices of the random lasing phase transition using , with finite size scaling techniques. The outcome of the scaling analysis is that the mode-locked random laser is in a mean-field universality class, though different from the mean-field class of the Random Energy Model and the glassy random laser in the narrow band approximation, that is, the fully connected version of the present model. The low temperature (high pumping) phase is finally characterized by means of the overlap distribution and evidence for the onset of replica symmetry breaking in the lasing regime is provided., 17 pages, 11 figures

We study a class of Hopfield models where the memories are represented by a mixture of Gaussian and binary variables and the neurons are Ising spins. We study the properties of this family of models as the relative weight of the two kinds of variables in the patterns varies. We quantitatively determine how the retrieval phase squeezes towards zero as the memory patterns contain a larger fraction of mismatched variables. As the memory is purely Gaussian retrieval is lost for any positive storage capacity. It is shown that this comes about because of the spherical symmetry of the free energy in the Gaussian case. Introducing two different memory pattern overlaps between spin configurations and each contribution to the pattern from the two kinds of variables one can observe that the Gaussian parts of the patterns act as a noise, making retrieval more difficult. The basins of attraction of the states, the accuracy of the retrieval and the storage capacity are studied by means of Monte Carlo numerical simulations. We uncover that even in the limit where the network capacity shrinks to zero, the (few) retrieval states maintain a large basin of attraction and large overlaps with the mismatched patterns. So the network can be used for retrieval, but with a very small capacity., 18 pages, 27 figures

In the field of disordered photonics, a common objective is to characterize optically opaque materials for controlling light delivery or performing imaging. Among various complex devices, multi-mode optical fibers stand out as cost-effective and easy-to-handle tools, making them attractive for several tasks. In this context, we leverage the reservoir computing paradigm to recast these fibers into random hardware projectors, transforming an input dataset into a higher dimensional speckled image set. The goal of our study is to demonstrate that using such randomized data for classification by training a single logistic regression layer improves accuracy compared to training on direct raw images. Interestingly, we found that the classification accuracy achieved using the reservoir is also higher than that obtained with the standard transmission matrix model, a widely accepted tool for describing light transmission through disordered devices. We find that the reason for such improved performance could be due to the fact that the hardware classifier operates in a flatter region of the loss landscape when trained on fiber data, which aligns with the current theory of deep neural networks. These findings strongly suggest that multi-mode fibers possess robust generalization properties, positioning them as promising tools for optically-assisted neural networks. With this study, in fact, we want to contribute to advancing the knowledge and practical utilization of these versatile instruments, which may play a significant role in shaping the future of machine learning.

In this chapter we report on the measurements of the overlap distribution of the replica symmetry breaking solution in complex disordered systems. After a general introduction to the problem of the experimental validation of the Parisi order parameter, we focus on the systems where the measurement has been possible for the first time: random lasers. Starting from first principles of light-matter interaction we sketch the main steps leading to the construction of the statistical mechanical model for the dynamics of light modes in a random laser, a spherical multi-p-spin model with complex spins. A new overlap is introduced, the intensity fluctuation overlap, whose probability distribution, under specific assumptions, is equivalent to the Parisi overlap distribution. The experimental protocol for measuring this overlap is based on the possibility of experimentally realizing real replicas. After a description of the first experiment on the random laser made of T5CO$_x$ grains we review and discuss various experiments measuring the overlap distribution, as well as the possible connection with Levy-like distribution of the intensity of the light modes around the laser threshold, the connection with turbulence in fiber lasers and the role of spatial etherogeneities of light modes in random media., Comment: Contribution to the edited volume "Spin Glass Theory and Far Beyond - Replica Symmetry Breaking after 40 years", World Scientific. 30 pages, 16 figures

The estimation of the transmission matrix of a disordered medium is a challenging problem in disordered photonics. Usually, its reconstruction relies on a complex inversion that aims at connecting a fully controlled input to the deterministic interference of the light field scrambled by the device. At the moment, iterative phase retrieval protocols provide the fastest reconstructing frameworks, converging in a few tens of iterations. Exploiting the knowledge of speckle correlations, we construct a new phase retrieval algorithm that reduces the computational cost to a single iteration. Besides being faster, our method is practical because it accepts fewer measurements than state-of-the-art protocols. Thanks to reducing computation time by one order of magnitude, our result can be a step forward toward real-time optical imaging that exploits disordered devices.

Recovering the transmission matrix of a disordered medium is a challenging problem in disordered photonics. Usually, its reconstruction relies on a complex inversion that aims at connecting a fully-controlled input to the deterministic interference of the light field scrambled by the device. At the moment, iterative phase-retrieval protocols provide the fastest reconstructing frameworks, converging in a few tens of iterations. Exploiting the knowledge of speckle correlations, we construct a new phase retrieval algorithm that reduces the computational cost to a single iteration. Besides being faster, our method is effective also using less measurements than state-of-the-art protocols. Thanks to reducing computation time by one order of magnitude, our result can be groundbreaking for real-time optical operations in medical imaging., Comment: 9 pages, 7 figures

The spherical $p$-spin is a fundamental model for glassy physics, thanks to its analytic solution achievable via the replica method. Unfortunately the replica method has some drawbacks: it is very hard to apply to diluted models and the assumptions beyond it are not immediately clear. Both drawbacks can be overcome by the use of the cavity method, which, however, needs to be applied with care to spherical models. Here we show how to write the cavity equations for spherical $p$-spin models on complete graphs, both in the Replica Symmetric (RS) ansatz (corresponding to Belief Propagation) and in the 1-step Replica Symmetry Breaking (1RSB) ansatz (corresponding to Survey Propagation). The cavity equations can be solved by a Gaussian (RS) and multivariate Gaussian (1RSB) ansatz for the distribution of the cavity fields. We compute the free energy in both ansatzes and check that the results are identical to the replica computation, predicting a phase transition to a 1RSB phase at low temperatures. The advantages of solving the model with the cavity method are many. The physical meaning of any ansatz for the cavity marginals is very clear. The cavity method works directly with the distribution of local quantities, which allows to generalize the method to dilute graphs. What we are presenting here is the first step towards the solution of the diluted version of the spherical $p$-spin model, which is a fundamental model in the theory of random lasers and interesting $per~se$ as an easier-to-simulate version of the classical fully-connected $p$-spin model., Comment: 26 pages

In ultra-fast multi-mode lasers, mode-locking is implemented by means of ad hoc devices, like saturable absorbers or modulators, allowing for very short pulses. This comes about because of nonlinear interactions induced among modes at different, well equispaced, frequencies. Theory predicts that the same locking of modes would occur in random lasers but, in absence of any device, its detection is unfeasible so far. Because of the general interest in the phenomenology and understanding of random lasers and, moreover, because it is a first example of self-starting mode-locking we devise and test a way to measure such peculiar non-linear coupling. Through a detailed analysis of multi-mode correlations we provide clear evidence for the occurrence of nonlinear mode-coupling in the cavity-less random laser made of a powder of GaAs crystals and its self-starting mode-locking nature. The behavior of multi-point correlations among intensity peaks is tested against the nonlinear frequency matching condition equivalent to the one underlying phase-locking in ordered ultrafast lasers. Non-trivially large multi-point correlations are clearly observed for spatially overlapping resonances and turn out to sensitively depend on the frequency matching being satisfied, eventually demonstrating the occurrence of non-linear mode-locked mode-coupling., 13 pages, 8 figures, 1 table

Linear problems appear in a variety of disciplines and their application for the transmission matrix recovery is one of the most stimulating challenges in biomedical imaging. Its knowledge turns any random media into an optical tool that can focus or transmit an image through disorder. Here, converting an input-output problem into a statistical mechanical formulation, we investigate how inference protocols can learn the transmission couplings by pseudolikelihood maximization. Bridging linear regression and thermodynamics let us propose an innovative framework to pursue the solution of the scattering-riddle.

The $XY$ model with four-body quenched disordered interactions and its discrete $p$-clock proxy is studied on bipartite random graphs by means of the cavity method. The phase diagrams are determined from the ordered case to the spin-glass case. Dynamic, spinodal and thermodynamic transition lines are identified by analyzing free energy, complexity and tree reconstruction functions as temperature and disorder are changed. The study of the convergence of the $p$-clock model to the $XY$ model is performed down to temperature low enough to determine all relevant transition points for different connectivity., 15 pages, 6 figures

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

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 figures

The Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field a la Imry-Ma we obtain a threshold value for the power $\rho$ of the long-range interaction, beyond which no critical behavior occurs. The critical threshold value is $\rho_c=3/2$, at a difference with the zero field model in which $\rho_c=2$. This prediction is confirmed by numerical computation of the ground states below, at, and above this threshold value. Some possible implications for the critical behavior of spin-glasses in a field are conjectured., Comment: 5 pages, 6 figures

An important prediction of Mode-Coupling-Theory (MCT) is the relationship between the power- law decay exponents in the {\beta} regime. In the original structural glass context this relationship follows from the MCT equations that are obtained making rather uncontrolled approximations and {\lambda} has to be treated like a tunable parameter. It is known that a certain class of mean-field spin-glass models is exactly described by MCT equations. In this context, the physical meaning of the so called parameter exponent {\lambda} has recently been unveiled, giving a method to compute it exactly in a static framework. In this paper we exploit this new technique to compute the critical slowing down exponents in a class of mean-field Ising spin-glass models including, as special cases, the Sherrington-Kirkpatrick model, the p-spin model and the Random Orthogonal model., Comment: 7 pages, 3 figures

Critical slowing down dynamics of supercooled glass-forming liquids is usually understood at the mean-field level in the framework of Mode Coupling Theory, providing a two-time relaxation scenario and power-law behaviors of the time correlation function at dynamic criticality. In this work we derive critical slowing down exponents of spin-glass models undergoing discontinuous transitions by computing their Gibbs free energy and connecting the dynamic behavior to static "in-state" properties. Both the spherical and Ising versions are considered and, in the simpler spherical case, a generalization to arbitrary schematic Mode Coupling kernels is presented. Comparison with dynamic results available in literature is performed. Analytical predictions for the Ising case are provided for any $p$., Comment: 9 pages

The Kovacs protocol, based on the temperature shift experiment originally conceived by A.J. Kovacs for glassy polymers, is implemented in an exactly solvable dynamical model. This model is characterized by interacting fast and slow modes represented respectively by spherical spins and harmonic oscillator variables. Due to this fundamental property, the model reproduces the characteristic non-monotonic evolution known as the ``Kovacs effect'', observed in polymers, in granular materials and models of molecular liquids, when similar experimental protocols are implemented., 8 pages, 6 figures

The study of the mean-field static solution of the Random Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched random magnetic interaction, is performed. The model exhibits a paramagnetic phase, described by a stable Replica Symmetric solution. When the temperature is decreased or the density increases, the system undergoes a phase transition to a Full Replica Symmetry Breaking spin-glass phase. The nature of the transition can be either of the second order (like in the Sherrington-Kirkpatrick model) or, at temperature below a given critical value, of the first order in the Ehrenfest sense, with a discontinuous jump of the order parameter and accompanied by a latent heat. In this last case coexistence of phases takes place. The thermodynamics is worked out in the Full Replica Symmetry Breaking scheme, and the relative Parisi equations are solved using a pseudo-spectral method down to zero temperature., 24 pages, 12 figures