The emerging of Inverse Freezing in spin glass models 

In recent years, the rapid advancement of Artificial Intelligence (AI) technologies has significantly influenced contemporary scientific research. This impact is redefining the scope of applied disciplines, while also stimulating theoretical interest in automated systems in a range of fields, including neuroscience, statistics, complex systems physics, engineering, and information theory. 

The statistical mechanics of spin glasses has traditionally functioned as a paradigmatic framework for modelling and interpreting a wide array of phenomena, extending from quantitative biology to computer science. 

In this talk, we explore models exhibiting inverse melting or inverse freezing, such as the Ghatak-Sherrington model and the disordered Blume-Emery-Griffiths-Capel model in the mean-field regime, relevant to both statistical mechanics and theoretical neuroscience. These models are revisited under a unified Hamiltonian framework that allows for a rigorous analytical treatment of replica symmetric and broken replica symmetric phases. The computations have been addressed through an adaptation and extension of Guerra's interpolations, showing perfect match with the results found via the well-known replica trick. 

This work may serve as a foundation for a series of studies aimed at understanding the role of inverse freezing phenomenon in neural networks, starting from the paradigmatic Hopfield model , offering an interpretative bridge between glassy phases in spin systems and memory degradation or overloading in neural networks. This interdisciplinary perspective sheds light on both the statistical mechanics of disordered systems and the theoretical foundations of learning and memory in artificial neural architectures. 

This research is inspired by the joint work with Adriano Barra and Emilio N.M. Cirillo (Sapienza University of Rome). 

Networks of neural networks: disentanglement of overlapping inputs 

When inputs are presented as superpositions—rather than as isolated patterns—recognition alone is not enough: a system must also be able to separate, or disentangle, the underlying constituents. This scenario is ubiquitous, from signal processing to representation learning, and it raises a natural theoretical question: which architectural and thermodynamic conditions allow an associative memory to decompose structured mixtures into their elementary components?

In this talk, we address this question within the statistical-mechanical theory of associative memories, focusing on a hetero-associative extension of the classic Hopfield framework. We present an extended Bidirectional Associative Memory (BAM) architecture capable of concurrently processing three or more patterns and show that, when organized as an ensemble of coupled BAM modules, the system exhibits emergent computational capabilities absent in a single network. Specifically, we design a layered associative Hebbian network that performs not only standard pattern recognition but also pattern disentanglement: when presented with a composite input—e.g., a musical chord—it can recover the individual elements composing it, namely the distinct notes.

Through a combined analytical and computational statistical-mechanics analysis, we derive the parameter regimes that enable successful disentanglement. Leveraging Guerra’s interpolation techniques and phase-diagram methods, we characterize the relevant phases and identify critical transitions associated with computational performance.

Numerical experiments confirm the model’s disentangling capability, with theoretical predictions closely matching empirical results. More broadly, this framework suggests a route toward a priori architectural tuning: by linking network design to the intrinsic organization of the data, it provides principled guidelines for optimizing deep learning pipelines in structured-mixture settings.

This work builds on joint research with Adriano Barra, Elena Agliari, Federico Ricci-Tersenghi (Sapienza University of Rome), and Martino Centonze (University of Bologna).

Quantum-gravity phenomenology and multi-messenger astrophysics

I offer a perspective on recent developments in quantum-gravity phenomenology attempting to exploit some opportunities provided by the advent of multi-messenger astrophysics.The relevant studies focus mainly on models of particle propagation in spacetimes with quantum properties and their implications for the observation of neutrinos and photons from the same astrophysical source. I also stress that this research program will soon require facing severe challenges for what concerns the collection, the storage and the analysis of very large amounts of data. 

Quantum correlations in elementary particle physics 

I will review recent work on entanglement and quantum correlations in elementary particle systems, ranging from neutrino oscillations to QED scattering processes. In both cases, the aim of our analysis is the search for yet unexplored features of fundamental interactions which can be unveiled by methods of quantum information. In our study, we make use of complete complementarity relations (CCR), which fully characterize both the local and nonlocal properties of a quantum system, whether pure or mixed.

For neutrinos, CCR applied to oscillations reveals a complex structure of correlations. Among these, the nonlocal advantage of quantum coherence emerges as a significant quantifier of coherence. I will also discuss chiral oscillations and their impact on spin quantum correlations in a lepton–antineutrino pair produced via weak interactions. Finally, I will address Leggett–Garg temporal inequalities in neutrino and meson oscillations.

I will then present recent results concerning the generation and distribution of (helicity) entanglement in Quantum Electrodynamics (QED) scattering processes. Viewing QED scatterings as quantum maps with distinct spectral properties reveals a unique role for maximal entanglement, which is strictly conserved in 2→2 scattering events involving only fermions. Moreover, repeated iterations of these maps inevitably lead to entanglement saturation, regardless of the initial state.

Interpolation Methods and Gaussian Equivalence Principles in Bayes-Optimal Deep Learning

We rigorously analyze fully-trained deep neural networks of arbitrary depth in the Bayes-optimal setting, focusing on the proportional scaling regime in which the number of training samples and the widths of the input and all hidden layers diverge proportionally. We prove an information-theoretic equivalence between the Bayesian deep neural network model trained on data generated by a teacher with matching architecture and a simpler model of optimal inference in a generalized linear model. This equivalence enables the computation of the optimal generalization error for deep neural networks in this regime. The proof is based on the interpolation method, pioneered by the late Francesco Guerra in the study of spin glasses together with F.L. Toninelli. This approach makes explicit all relevant correlations within the model, providing a conceptually smooth and self-contained derivation of the so-called deep Gaussian equivalence principle, conjectured in Cui et al. (2023, arXiv:2302.00375).

Based on "Information-theoretic reduction of deep neural networks to linear models in the overparametrized proportional regime”, F. Camilli, D. Tieplova, E. Bergamin, J. Barbier, 38th Annual Conference on Learning Theory (COLT 2025)

From associative memory to generative modelling: the Restricted Boltzmann Machine

The Restricted Boltzmann Machine is a bipartite model inspired by statistical physics, in which a layer of visible variables interacts with a set of latent units. When the hidden units have a Gaussian prior, the model is equivalent to a Hopfield network and therefore displays the familiar properties of associative memory. However, in this form it is a rather poor generative model, as it cannot capture higher-order statistical correlations.

If one instead uses Bernoulli hidden units, the situation is reversed: the model becomes a very powerful generative model, capable of learning complex statistical structures. Yet, in this case, associative memory seems to disappear, which seems inconsistent given what is observed in empirical set-up.

In this talk, I will show how this apparent contradiction can be resolved by viewing patterns as extensive combinations of many Bernoulli units. From this perspective, the phase diagram of the model is changed and the associative memory properties restore. I will conclude by presenting numerical evidence showing that these phase transitions do occur under realistic learning conditions.

TBA

Self-Sustained Oscillations in Natural Systems: some examples

Complex environmental systems frequently display persistent oscillatory. Such dynamics are characteristic of self-oscillating nonlinear systems, where energy input, dissipation, and coupling mechanisms lead to coherent temporal organization. Three distinct environmental contexts in Italy, the sea-level variability in the Venice Lagoon, the hydrothermal seismic noise at Ischia Island, and the acoustic soundscape of the Trevi Fountain in Rome, within a unified framework of nonlinear dynamical systems and complexity theory are investigated.

Despite the diversity of physical settings, all three systems exhibit signatures of emergent coherence, nonlinearity, and sustained oscillations. In the Venice Lagoon, the appearance of higher-order tidal constituents and amplitude inversion between modes points to a globally organized self-oscillatory regime. At Ischia, the continuous seismic “whisper” generated by the hydrothermal system is consistent with a network of coupled resonators, where mode activation depends on energy thresholds and nonlinear feedback. In Rome, the analysis of the Trevi Fountain soundscape during the exceptional quiet conditions of the COVID-19 lockdown revealed a stable and persistent acoustic signature associated with the fountain. Under ordinary conditions, the fountain sound cannot be fully separated from human vocal noise, indicating that the soundscape is not a simple linear superposition of independent sources. Rather, the fountain acts as a structuring element of the acoustic environment, influencing collective human behavior. Together, these observations support a unifying interpretation of diverse environmental phenomena as manifestations of self-organized, nonlinear oscillatory dynamics, highlighting the relevance of complex systems theory for understanding environmental variability across fluid, solid, and acoustic domains.

Implementing the Einstein Classical Program

Abstract

Memories on Francesco Guerra and his novel approach to constuctive field theory

Two results on the Hopfield model 

I will present two results on the Hopfield model at low load, one for high temperature and one for low temperature. The first one concerns a log-Sobolev inequality in a region of the phase diagram that coincide with the high-temperature region at low load. The second result is revisiting the localisation around the Mattis states of the Gibbs measure, generalising previous theorems of Bovier-Gayrard and Talagrand. 

Generalization performance of narrow one-hidden layer networks in the teacher-student setting 

Understanding the generalization abilities of neural networks for simple input-output distributions is crucial to account for their learning performance on real datasets. The classical teacher-student setting, where a network is trained from data obtained thanks to a label-generating teacher model, serves as a perfect theoretical test bed. In this work, we give a complete theoretical account of the teacher-student generalization performance of fully connected one-hidden layer networks with generic activation functions, in a regime where the number of hidden units is large, yet much smaller than the input dimension. Using methods from statistical physics, we provide closed-form expressions for the typical performance of both finite temperature (Bayesian) and empirical risk minimization estimators. In doing so, we highlight the presence of a transition where hidden neurons specialize when the number of samples is sufficiently large and proportional to the number of parameters of the network. Our theory accurately predicts the generalization error of neural networks trained on regression or classification tasks with either noisy full-batch gradient descent (Langevin dynamics) or full-batch gradient descent. 

Quadratic unconstrained binary optimization with random coefficients and the SK lattice gas

Quadratic Unconstrained Binary Optimization (QUBO or UBQP) is concerned with maximizing/minimizing a quadratic form in Boolean variables.

Random instances of QUBO are used to test the effectiveness of heuristic algorithms. In the statistical mechanics literature, QUBO is a lattice gas counterpart to the (generalized) Sherrington–Kirkpatrick spin glass model. In QUBO with random independent coefficients with zero mean and finite variance the minimum and maximum per particle do not depend on the details of the distribution and are concentrated around their expected values. With the help of numerical simulations, we also study the minimum and maximum of the objective function and provide some insight into the structure of the minimizer and the maximizer. We argue that also this structure is rather robust. Our findings hold also in the diluted case where each of the couplings is allowed to be zero with probability going to 1 as  in a suitable way. 

Spurious overlaps in attractor networks

In attractor neural networks, spurious overlaps are detrimental correlations between the

network’s state and memories that are distinct from the memory that is being retrieved.

We show that while individual spurious overlaps are vanishingly small in the large net-

work size limit, their collective effect can suppress the mean input currents to neurons,

promoting sparse neural activity and as a result increasing storage capacity. While clas-

sical attractor models do not exhibit this behavior, plasticity rules inferred from neuro-

physiological experiments do. These findings offer new insights into memory storage in

biologically inspired networks. 

Non-classical hydrodynamics and particle systems

I will  discuss the  formal derivation as hydrodynamic limit, under diffusive scaling when the density and temperature at initial time have gradients of order 1,  of  completely new equations for  a Hamiltonian particle system in the time-dependent setting. The limiting equations    are different  from the usual Navier-Stokes-Fourier eqs and cannot be predicted by the classical fluid theory. Analogous eqs can be derived starting from the Boltzmann equation.

General properties of  the multiscale measures 

We review the definition and basic properties of a class of probability measures used to describe physical systems in a peculiar out-of-equilibrium regime and appearing in Guerra’s scheme for the study of mean-field spin glasses. These measures are characterized by a hierarchical structure and admit a clear thermodynamic interpretation. We also discuss a recent interpretation of these measures from an information-theoretic perspective. 

Francesco Guerra and the History of Physics: the book “Vita di Ettore Majorana”

Francesco Guerra’s research in the history of physics is traced from its beginnings in 2002 to his final years, culminating in the book “Vita di Ettore Majorana,” which we co-authored. Unfortunately, he saw it only in draft form.

This work represents one of Francesco’s most significant contributions to the history of twentieth-century physics. Based on a broad and rigorous analysis of archival materials, the book transcends the stereotypical image of Majorana as a genius lacking common sense. Instead, it presents a scientist who was certainly brilliant, but also attentive to the practical aspects of life, and not at all misanthropic or antisocial as the cliché suggests. Majorana was attentive to his academic career, sought to uphold the priority of his discoveries, was confident in himself and his ideas, stood up to figures such as Enrico Fermi, Paul Dirac, and Werner Heisenberg, counterattacked when necessary, and appeared fully integrated into the international scientific community, while remaining completely detached from the milieu of Via Panisperna.

Finally, regarding the fate of Ettore Majorana, new documents discovered during the research allow us to place his death in September 1939 and to conclude that the alleged sightings of Majorana in Argentina, Venezuela, or elsewhere, as well as his supposed presence in Germany during the war, are completely unfounded.

Scholia in Guerra-Toninelli

The legacy of a Teacher Boltzmann machine.

Some teachers offer countless suggestions, while others leave an indelible mark with only a few words. Some students resonate perfectly with their teacher, while others  learn in different forms. Some teachings shine with clarity and directness, while others veil a subtler truth beneath the surface.

Boltzmann Machines behave in much the same way when they learn. I will revisit some results concerning the learning capabilities of Boltzmann Machines within the teacher–student framework. We explore the role played by the complexity of the  teacher machine,  the “temperament”  of the student machine, as well as the properties of the dataset, in terms of  size and informational content.

Photonic realization of a Hopfield associative memory by multiphoton entanglement and interference

In this talk, we will discuss a recently discovered connection [1] between multiphoton quantum interference, a key component of emerging photonic quantum technologies, and Hopfield-like Hamiltonian models associated with classical neural networks, which are paradigmatic models of associative memory and machine learning in systems of artificial intelligence. A major challenge in this context is that simulating the dynamics of fully connected networks with multi-neuronal synaptic couplings requires a computational time that scales super-extensively with the number of neurons. 

Analog photonic computation offers a promising path to drastically reduce the simulation time, as pointed out in a previous study [2] in which coherent classical light was used to simulate a 2-body Hopfield model. In order to generalize this correspondence to the multi-synaptic, large-storage regime, an additional mechanism is required in the proposed photonic architecture: the use of entangled photons and interferometry.

Specifically, we demonstrate that a system composed of N_p indistinguishable photons in superposition over M field modes, combined with a controlled array of M binary phase-shifters and a linear-optical interferometer, produces output photon statistics that can be described by a p-body Hopfield Hamiltonian. This Hamiltonian involves M Ising-like neurons (with states ±1), where the interaction order is given by p = 2N_p.

Hopfield models are characterized by two key parameters: the temperature T, and the storage load α, defined as the number of stored patterns per neuron. These models exhibit three distinct phases: the retrieval phase (low T, low α), the glassy phase (low T, high α), and the paramagnetic phase, which emerges at sufficiently high T. In our proposed photonic implementation, both of these parameters can be experimentally controlled, enabling the exploration of the different phases of the system.

As an illustrative example, we discuss the generalized 4-body Hopfield model arising from our design. We show that it undergoes a transition from a memory retrieval phase to a memory blackout (spin-glass) regime as the number of stored patterns increases.

The proposed mapping provides a new avenue toward the implementation and investigation of disordered and complex classical systems via efficient photonic quantum simulators. Conversely, it allows for the interpretation of structured photonic systems through the lens of classical spin Hamiltonians.

References:

[1] G. Zanfardino, S. Paesani, L. Leuzzi, R. Santagati, F. Illuminati, G. Ruocco, and M. Leonetti, “Multiphoton quantum simulation of the generalized Hopfield memory model”,  doi.org/10.48550/arXiv.2504.00111 (Doi of the proposed work).

[2]  M. Leonetti, E. Hormann, L. Leuzzi, G. Parisi, and G. Ruocco, “Optical computation of a spin glass dynamics with tuneable complexity”, Proceedings of the National Academy of Sciences 118, e2015207118 (2021).