arXiv subjects

Luca Leuzzi

Publications and source records attributed to Luca Leuzzi.

At least 19 recordsLinked to original sources

arXiv2026

Quenched complexity of marginal states in the Sherrington--Kirkpatrick spin glass

In the Sherrington-Kirkpatrick model the exponentially many metastable states are marginal, so counting them requires breaking the BRST supersymmetry or a two-group replica Ansatz. For four decades this complexity was known only in the annealed approximation, which is unstable at low free energy and predicts states below the Parisi equilibrium free energy. We compute it quenched, with one step of replica symmetry breaking. It leaves the annealed curve where it becomes unstable and vanishes next to the full-RSB equilibrium free energy: the lowest marginal states are thus plausibly the equilibrium states themselves.

cond-mat.dis-nn↗

arXiv2026

Aging Phase Diagram and Exact Asymptotic Energies of Mixed Spherical Spin Glasses

We determine the aging phase diagram of mixed spherical $(p+s)$-spin glasses quenched from random configurations to zero temperature. Solving the asymptotic dynamical equations of Cugliandolo and Kurchan, we find transitions between aging states with one, two, and continuously many effective temperatures, including phases in which discrete and continuous hierarchies coexist. We obtain analytic expressions for the energies reached asymptotically by the dynamics. In models combining pairwise and higher-order interactions, and sufficiently separated interaction orders ($s>3$), a fully continuous phase exists, where the gradient descent relaxation converges to the algorithmic energy lower bound. Instead, in mixed models with only high-order interactions ($p\ge3$) the energy relaxation remains strictly above it. Numerical integration of the dynamical equations is consistent with the predicted two-step and mixed discrete--continuous aging states.

cond-mat.dis-nn↗

arXiv2026

Observation of associative-memory retrieval and spin-glass phases on a photonic quantum simulator

Models of interacting complex systems provide the fundamental statistical physics reference frame for the study and the understanding of associative memories, machine learning, and the dynamics of neural networks. On the other hand, simulating complex multi-synaptic interactions on a classical hardware is computationally demanding due to the super-linear scaling of the system complexity. Photonic quantum technologies provide a promising solution to these limitations by leveraging on their inherent speed and parallel processing ability in order to simulate complex networks. Recently, a connection between multiphoton processes and generalized $p$-body Hopfield models has been theoretically established. Here, we design and demonstrate an experimental platform that exploits single photons distributed across a set of optical modes, in which controlled arrays of binary phase shifters act as Ising-like neurons. We focus specifically on a fully connected Hopfield Hamiltonian with four-body local interaction terms, realized via two-photon processes. Through quantum simulations on programmable photonic processors, the study identifies three distinct regimes: a memory retrieval phase, a spin-glass memory "black-out" phase, and a paramagnetic phase. Experimental results confirm successful memory retrieval at low storage capacities and temperatures, where the system consistently relaxes to fixed points with high memory overlap, effectively reconstructing the stored patterns. Future research will extend the platform design to investigate networks with local or dilute interactions, while advances in the realization of scalable photonic circuits will enable architectures that encompass very large numbers of interacting spins.

quant-ph↗

arXiv2026

DYNAMITE: A high-performance framework for solving Dynamical Mean-Field Equations

Understanding the dynamics of systems evolving in complex and rugged energy landscapes is central across physics, economics, biology, and computer science. Disordered mean-field models provide a powerful framework, as exact Dynamical Mean-Field Equations (DMFE) can be derived. However, solving the DMFE -- a set of coupled integral-differential equations for two-time functions -- remains a major numerical challenge. So far, large-time solutions of DMFE rely either on analytical approaches, such as the Cugliandolo--Kurchan ansatz based on assumptions like weak ergodicity breaking (which is known to fail in some cases), or on numerical integrations that reliably reach times $O(10^3)$ and extend further only via poorly controlled approximations. Consequently, no general method currently exists to solve DMFE at very long times, limiting the study of slow dynamics in complex landscapes. We present \textsc{Dynamite} (DYNAmical Mean-fIeld Time Evolution solver), a high-performance framework for solving DMFE up to unprecedented times $t=O(10^7)$. It combines non-uniform interpolation, adaptive time stepping, and numerical `renormalization' of memory, enabling accurate evaluation of history integrals. Its asymptotic runtime is linear, with sublinear memory scaling. Stability and precision are ensured via an adaptive Runge--Kutta scheme and periodic sparsification of the past. \textsc{Dynamite} achieves orders-of-magnitude speedups over uniform-grid methods while maintaining accuracy and reproducibility on CPU and GPU architectures. Benchmarks on glassy mean-field models, including the mixed spherical $p$-spin system, demonstrate access to aging and relaxation regimes previously out of reach. The framework provides a reproducible and extensible foundation for studying long-memory dynamical systems.

cond-mat.dis-nn↗

arXiv2025

Smoothed-Cubic Spin-Glass Model of Random Lasers

We study the equilibrium glassy behavior of a multimode random laser model with nonlinear four-body quenched disordered interactions and a global smoothed-cubic constraint on mode intensities. This constraint, which provides a more realistic representation of gain saturation than the commonly used spherical constraint, prevents intensity condensation while preserving the dense, long-range interaction structure characteristic of many multistate random lasers. The model effective Hamiltonian is a function of mode amplitudes with random frequencies and is defined on a complete mode-locked graph. Using large-scale GPU-accelerated Monte Carlo simulations with the Parallel Tempering algorithm, we analyze systems of varying sizes to probe their thermodynamic-limit behavior. Finite-size scaling of the specific heat, of the Parisi overlap distributions, and of the inverse participation ratio's reveals a spin-glass transition, with critical exponents matching the mean-field Random Energy Model universality class. The smoothed-cubic constraint produces broad, non-condensed intensity distributions, avoiding the pseudo-condensation seen in spherical models on the same interaction graph. Our results show that more realistic gain-saturation constraints preserve spin-glass characteristics while enabling simulations of larger, more dilute systems, providing a robust framework for studying glassy random lasers with self-starting mode-locking.

cond-mat.dis-nn↗

arXiv2025

Learning and Testing Inverse Statistical Problems For Interacting Systems Undergoing Phase Transition

Inverse problems arise in situations where data is available, but the underlying model is not. It can therefore be necessary to infer the parameters of the latter starting from the former. Statistical mechanics offers a toolbox of techniques to address this challenge. In this work, we illustrate three of the main methods: the Maximum Likelihood, Maximum Pseudo-Likelihood, and Mean-Field approaches. We begin with a thorough theoretical introduction to these methods, followed by their application to inference in several well-known statistical physics systems undergoing phase transitions. Namely, we consider the ordered and disordered Ising models, the vector Potts model, and the Blume-Capel model on both regular lattices and random graphs. This discussion is accompanied by a GitHub repository that allows users to both reproduce the results and experiment with new systems.

cond-mat.stat-mech↗

arXiv2025

Multiphoton quantum simulation of the generalized Hopfield memory model

In the present work, we introduce, develop, and investigate a connection between multiphoton quantum interference, a core element of emerging photonic quantum technologies, and Hopfieldlike Hamiltonians of classical neural networks, the paradigmatic models for associative memory and machine learning in systems of artificial intelligence. Specifically, we show that combining a system composed of Nph indistinguishable photons in superposition over M field modes, a controlled array of M binary phase-shifters, and a linear-optical interferometer, yields output photon statistics described by means of a p-body Hopfield Hamiltonian of M Ising-like neurons +-1, with p = 2Nph. We investigate in detail the generalized 4-body Hopfield model obtained through this procedure and show that it realizes a transition from a memory retrieval to a memory black-out regime, i.e. a spin-glass phase, as the amount of stored memory increases. The mapping enables novel routes to the realization and investigation of disordered and complex classical systems via efficient photonic quantum simulators, as well as the description of aspects of structured photonic systems in terms of classical spin Hamiltonians.

quant-ph↗

arXiv2024

Logarithmic critical slowing down in complex systems: from statics to dynamics

We consider second-order phase transitions in which the order parameter is a replicated overlap matrix. We focus on a tricritical point that occurs in a variety of mean-field models and that, more generically, describes higher order liquid-liquid or liquid-glass transitions. We show that the static replicated theory implies slowing down with a logarithmic decay in time. The dynamical equations turn out to be those predicted by schematic Mode Coupling Theory for supercooled viscous liquids at a $A_3$ singularity, where the parameter exponent is $\lambda=1$. We obtain a quantitative expression for the parameter $\mu$ of the logarithmic decay in terms of cumulants of the overlap, which are physically observable in experiments or numerical simulations.

cond-mat.dis-nn↗

arXiv2022

Intensity pseudo-localized phase in the glassy random laser

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.

cond-mat.dis-nn↗

arXiv2022

Hopfield-like open channel retrieval for disordered optical media

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.

physics.optics↗

arXiv2022

Low-power multi-mode fiber projector overcomes shallow neural networks classifiers

In the domain of disordered photonics, the characterization of optically opaque materials for light manipulation and imaging is a primary aim. 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 cast 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 is higher than that obtained with the standard transmission matrix model, a widely accepted tool for describing light transmission through disordered devices. We conjecture 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 suggest that the class of random projections operated by multi-mode fibers generalize better to previously unseen data, 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 neuromorphic machine learning.

physics.optics↗

arXiv2022

Universality class of the mode-locked glassy random laser

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.

cond-mat.dis-nn↗

arXiv2022

Replica symmetry breaking in random lasers: experimental measurement of the overlap distribution

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.

cond-mat.dis-nn↗

arXiv2022

A generalized Hopfield model to store and retrieve mismatched memory patterns

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.

cond-mat.dis-nn↗

arXiv2022

Worldwide bilateral geopolitical interactions network inferred from national disciplinary profiles

A disciplinary profile of a country is defined as the versor whose components are the numbers of articles produced in a given discipline divided the overall production of the country. Starting from the Essential Science Indicators (ESI) schema of classification of subject area, we obtained the yearly disciplinary profiles of a worldwide graph, where on each node sits a country, in the two time intervals [1980-1988] and [1992-2017], the fall of the Berlin Wall being the watershed. We analyse the empirical pairwise cross-correlation matrices of the time series of disciplinary profiles. The contrast with random matrix theory proves that, beyond measurement noise, the empirical cross-correlation matrices bring genuine information. Arising from the Shannon theorem as the least-structured model consistent with the measured pairwise correlations, the stationary probability distribution of disciplinary profiles can be described by a Boltzmann distribution related to a generalized $n_d$-dimensional Heisenberg model. The set of network interactions of the Heisenberg model have been inferred and to it they have been applied two clusterization methods, hierarchical clustering and principal component analysis. On a geopolitical plane this allow to obtain a characterization of the worldwide bilateral interactions based on physical modeling. A simple geopolitical analysis reveals the consistency of the results obtained and gives a boost to deeper historical analysis. In order to obtain the optimal set of pairwise interactions we used a Pseudo-Likelihood approach. We analytically computed the Pseudo-Likelihood and its gradient. The analytical computations deserve interest in whatever inference Bayesian problem involving a $n_d$-dimensional Heisenberg model.

physics.soc-ph↗

arXiv2021

Single-step transmission matrix retrieval for fast imaging through multi-mode fibers

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.

physics.comp-ph↗

arXiv2020

Solving the fully-connected spherical $p$-spin model with the cavity method: equivalence with the replica results

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.

cond-mat.stat-mech↗

arXiv2020

Spin glasses in a field show a phase transition varying the distance among real replicas (and how to exploit it to find the critical line in a field)

We discuss a phase transition in spin glass models which have been rarely considered in the past, namely the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e. at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that even in a paramagnetic phase the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite-dimensional spin glasses. This is a notoriously difficult computational problem, because of huge finite-size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, $h>0$ and $T_c(h)<T<T_c(h=0)$, of the $d=1$ spin glass model with long-range interactions decaying fast enough to be outside the regime of validity of the mean-field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.

cond-mat.dis-nn↗