The Department of Physics at Sapienza University of Rome stands as a world-renowned epicenter for the study of complex systems. The research activities carried on represent the vanguard of statistical mechanics, driven by the profound paradigm shift initiated by the theory of Replica Symmetry Breaking (RSB). Originally formulated by Giorgio Parisi to solve the enigmatic behavior of spin glasses, RSB has evolved far beyond its condensed matter origins. It now serves as the universal mathematical language for understanding macroscopic systems characterized by disorder, frustration, and rugged energy landscapes featuring exponentially many metastable states.

The awarding of the 2021 Nobel Prize in Physics to Giorgio Parisi definitively validated this framework's foundational importance to modern science. At this Excellence Research Center, a deeply collaborative group of researchers—spanning faculty from Sapienza and the National Research Council (CNR)—continues to push the boundaries of this field. Their collective work demonstrates how the principles governing magnetic alloys can be seamlessly translated to solve critical problems across diverse disciplines.

In recent years, the group's focus has naturally expanded into the theoretical foundations of computer science, biology, and machine learning. A complex system, whether it is a rapidly quenched liquid forming a glass, a folded protein, a heavily constrained optimization problem, or a deep neural network training on vast datasets, shares a fundamental topological feature: a highly complex configuration space. The group’s recent publications emphasize a multidisciplinary approach, utilizing advanced analytical techniques (such as the cavity method and replica theory) and cutting-edge numerical simulations (like parallel tempering and large-scale Monte Carlo) to map these rugged landscapes.

The summary below details ten core subjects that define the group’s current scientific output, followed by profiles of the key faculty members who lead this extraordinary intellectual endeavor.

Spin Glasses
Spin glasses represent the archetypal complex system and remain a central pillar of the group's research. These disordered magnetic systems, characterized by conflicting ferromagnetic and antiferromagnetic interactions (frustration), possess a free-energy landscape littered with numerous degenerate equilibrium states. Recent work by this collaboration has focused heavily on the behavior of spin glasses in finite dimensions and the effects of external magnetic fields. Over the last three years, the group has produced critical numerical and analytical evidence investigating the validity of the de Almeida-Thouless (dAT) line in three-dimensional systems, a long-standing debate in the community. Furthermore, advanced diagrammatic expansions and loop calculations, combined with state-of-the-art simulations, have provided new estimates on the critical exponents of the spin glass phase, solidifying the applicability of infinite-dimensional RSB theory to real-world, finite-dimensional materials.
 

Structural Glasses
Understanding the glass transition—how a liquid ceases to flow and becomes an amorphous solid without a distinct structural phase change—is one of the most profound open questions in condensed matter physics. They have been instrumental in bridging the gap between liquid-state theory and the statistical physics of disordered systems. Building upon the exact solution of structural glasses in the limit of infinite dimensions, recent research has explored the "Gardner transition," a secondary phase transition deep within the glassy state where the amorphous solid yields to an even more complex, marginally stable hierarchy of states. Recent publications have extended these infinite-dimensional insights to lower, physical dimensions, utilizing mode-coupling theory and replica methods to provide a unified framework that explains not only structural glasses but also the jamming transition in granular materials.
 

Disorder in Materials and Random Lasers
The principles of disordered statistical mechanics have profound implications for photonics and wave propagation. The group has explored complex optical systems, most notably random lasers. Unlike traditional lasers that rely on highly structured mirrors to form an optical cavity, random lasers achieve coherent light emission through multiple scattering events within a disordered dielectric medium. Their recent contributions involve mapping the nonlinear wave interactions of random lasers onto statistical mechanical spin models (such as phase-coupled oscillators and spherical spins). By applying replica theory to these models, the group has successfully predicted mode-locking regimes and emission spectra transitions, effectively treating the sudden onset of coherent narrow-band emission in random media as a thermodynamic phase transition into a glassy state of light.

Anomalous Diffusion
While classical Brownian motion describes diffusion in simple fluids, transport inside complex, crowded environments—such as the interior of living cells, porous media, or active matter systems—often exhibits anomalous behavior. They have conducted extensive research into single-file diffusion and tracer dynamics in deeply restricted geometries. Their recent work focuses on non-Gaussian fluctuations and extreme value statistics in non-equilibrium systems. By utilizing large-scale stochastic simulations and exact analytical derivations, they have uncovered universal scaling laws governing how macroscopic observables fluctuate when particles are subjected to severe volume exclusion and temporal memory effects, providing crucial insights for soft matter physics and biophysics.
 

Protein Evolutionary Dynamics
The group fundamentally advanced the understanding of protein evolution by mapping the complex topology of fitness landscapes through the lens of epistasis. By applying generative models like Boltzmann Machines and generalized fitness mappings, their work demonstrates how the context-dependence of mutational effects governs the speed and predictability of evolutionary paths. A key breakthrough involves the use of disordered systems theory to identify the proper correlation functions—connected four-point space-time correlations—required to accurately capture the statistical structure of sequence space and its impact on evolutionary dynamics. This framework reveals that amino acid evolution transitions through different dynamical regimes, where collective epistasis creates distinct time scales that separate fast mutational shifts from the slow, entrenched evolution of the protein’s scaffold. Ultimately, these predictive tools quantify how functional bottlenecks and mutational contingencies determine the long-term resilience and divergence of protein families.
 

Ecologies
Ecosystems are highly interconnected networks of interacting species, where the survival of one depends heavily on its complex relationships (predation, competition, mutualism) with others. They have adapted the physics of disordered systems to macroscopic biology, fundamentally advancing theoretical ecology. Using generalized Lotka-Volterra equations with random interaction matrices, their recent research maps the phase diagrams of large ecological networks. The work demonstrates how ecosystems transition from stable, highly diverse states to chaotic, dynamically fluctuating regimes, or suffer cascading extinctions, depending on the variance of species interactions. This application of replica theory provides vital predictive tools for understanding biodiversity stability, species packing, and the resilience of ecological communities against external shocks.
 

Neural Networks, Hopfield Models, Dreaming
The intersection of statistical mechanics and neuroscience has deep historical roots in Rome (also related to the late and never forgotten Daniel Amit). The group continues to lead research on dense associative memories and Hopfield networks. A fascinating recent avenue of research involves the statistical mechanics of "unlearning" or "dreaming" in neural networks. Inspired by biological sleep cycles, the group has developed models where networks undergo offline phases of spontaneous activity to prune spurious local minima (unwanted mixed memory states). They have, among other results, published theoretical breakthroughs showing how this REM-like (Rapid Eye Movement) unlearning phase dramatically increases the storage capacity of the network and improves the basin of attraction for robust memory retrieval, formally linking synaptic plasticity rules to thermodynamic optimization.
 

Optimization and Search of Optimal Solutions
Many fundamental problems in computer science—such as graph coloring, the Traveling Salesperson Problem, and random k-satisfiability (k-SAT)—are classified as NP-hard. These constraint satisfaction problems exhibit a striking phase transition: as the ratio of constraints to variables increases, the problems suddenly shift from easy to solve to virtually impossible. They have utilized the cavity method to understand the algorithmic thresholds of these problems. Their recent work maps the "clustering transition," where the space of valid solutions shatters into isolated, disconnected components, trapping local search algorithms. By understanding the geometric structure of this shattered landscape, they have developed sophisticated message-passing algorithms (like Survey Propagation) that can find optimal or near-optimal solutions in regimes where classical algorithms completely fail.
 

Machine Learning  and AI
The astonishing empirical success of Deep Learning has outpaced its theoretical understanding. Why do heavily overparameterized neural networks, which possess loss landscapes with exponentially many local minima, consistently converge to solutions that generalize well to unseen data? They have leveraged their expertise in the geometry of high-dimensional landscapes to answer this. Recent contributions apply the theory of jamming and the replica method to the loss surfaces of generative models and deep classifiers. They have provided analytical frameworks demonstrating how overparameterization smooths the landscape, effectively transforming isolated glassy minima into connected manifolds of zero loss. This statistical mechanical perspective is crucial for establishing a rigorous theoretical foundation for modern Artificial Intelligence.
 

Diffusion Models and Generative AI
Diffusion models currently represent the state-of-the-art in generative AI (e.g., generating high-fidelity images and audio). These models operate by gradually corrupting data with Gaussian noise until it is entirely structureless, and then learning to reverse this stochastic process. The group has approached diffusion models through the lens of non-equilibrium statistical mechanics and stochastic thermodynamics. Their recent work formally maps the training of score-based models to the entropy production and Langevin dynamics of physical systems relaxing in a potential. By analyzing the continuous-time dynamics of these networks, they have identified fundamental theoretical limits on generation speed and proposed novel sampling algorithms that drastically reduce computational cost without sacrificing the fidelity of the generated data.

Faculty Members and Core Contributors
The exceptional output of this group is driven by a highly integrated team of scientists, featuring faculty from Sapienza University and resident researchers from the National Research Council (CNR). Their diverse expertise creates a unique, synergetic environment.

Giorgio Parisi - Nobel Laureate
Emeritus Professor and 2021 Nobel Laureate in Physics. His invention of the Replica Symmetry Breaking theory revolutionized the study of complex systems. Though Emeritus, he remains exceptionally active, continuously providing visionary insights that connect abstract mathematical physics to concrete phenomena in biology, materials science, and computing. His presence ensures the center remains a global beacon for theoretical physics.

Chiara Cammarota - Associate Professor
Chiara Cammarota is a leading expert in applying statistical mechanics to high-dimensional complex systems. She pioneered the translation of disordered energy landscape theory—originally developed for structural glasses—into diverse fields such as theoretical ecology, artificial intelligence, and societal phenomena. By leveraging the Kac-Rice formula, the replica method, and random matrix theory, her research provides a rigorous mathematical framework for understanding opinion dynamics, population dynamics of large ecosystems, and learning processes in inference problems and machine learning.

Luca Leuzzi - Research Director (CNR)
Operating at the intersection of statistical mechanics and optics, Luca Leuzzi is an expert in the statistical physics of random and complex photonics. His groundbreaking work applies the replica trick and spin-glass formalisms to understand random lasers, optical wave turbulence, and nonlinear wave propagation in disordered media. His dual expertise in advanced analytical methods and large-scale numerical simulations makes him a vital asset to the group.

Enzo Marinari - Full Professor
He is an expert in computational statistical mechanics. He was instrumental in the development of simulated tempering and parallel exchange Monte Carlo methods, which are now standard tools globally for simulating systems with rugged energy landscapes. His recent focus spans across anomalous diffusion, the rigorous bounds of finite-dimensional spin glasses, and the statistical physics of neural networks and machine learning architectures.

Federico Ricci-Tersenghi - Full Professor
He bridges theoretical physics and computer science. He is an expert on constraint satisfaction problems, phase transitions in randomized algorithms, and the statistical physics of machine learning. His work on message-passing algorithms and his recent deep dives into the theoretical underpinnings of Generative AI and Diffusion models place him at the forefront of the physics of computation.

Tommaso Rizzo - Senior Researcher (CNR)
He is an expert in advanced theoretical techniques, particularly diagrammatic expansions, loop corrections, and field theories for disordered systems. His technical competence has been crucial in calculating critical properties of spin glasses and structural glasses in finite dimensions. His theoretical frameworks often provide the necessary analytical bedrock upon which complex numerical simulations are validated.

Francesco Zamponi - Associate Professor
He has had a crucial role in developing the exact theory of the glass transition in infinite dimensions. He contributed to connecting the abstract replica theory to tangible, experimental observables, such as the jamming transition in granular matter and the evolutionary dynamics of proteins. More recently, he has made significant strides in applying these exact theoretical frameworks to map the loss landscapes of modern Artificial Intelligence networks.