The study of disordered systems plays a central role in modern statistical physics and complex systems science, with applications ranging from condensed matter physics and materials science to optimization, machine learning, information theory, and biological systems. Randomness, frustration, and collective phenomena generate rich emergent behaviors that cannot be understood within the framework of ordered equilibrium systems alone. The understanding of these phenomena requires both advanced theoretical tools and the ability to connect abstract models with experimentally observable properties and computational problems. Within this context, the course on Disordered Systems aims to provide the fundamental theoretical methods and conceptual tools necessary for the description of complex systems characterized by quenched disorder, frustration, and nontrivial collective dynamics. Particular emphasis is placed on the interplay between statistical mechanics, phase transitions, symmetry breaking, and probabilistic methods. The course introduces the theoretical framework of spin glasses and random systems, discussing both phenomenological and microscopic descriptions, together with the mathematical methods required for their analysis, including mean-field approximations, replica methods, cavity approaches, and effective Hamiltonian techniques. The course also addresses stochastic growth processes and nonequilibrium phenomena, with special attention to surface growth models and the Kardar-Parisi-Zhang universality class, highlighting the connections between statistical physics, stochastic differential equations, and random media. In addition, the course introduces modern approaches to optimization problems on sparse random graphs, discussing message-passing algorithms such as belief propagation, warning propagation, and survey propagation, as well as their relation to replica symmetry breaking and constraint satisfaction problems. Alongside the theoretical developments, illustrative examples and applications are discussed throughout the course in order to connect the formalism with physical phenomena and computational problems of current scientific interest. Particular emphasis is placed on the emergence of collective behavior from disorder, on the role of scaling and universality, and on the interdisciplinary relevance of disordered systems methods across physics, computer science, and complex systems research.

At the end of the course, students are expected to: * Know and understand the fundamental concepts of equilibrium statistical mechanics, including the Boltzmann distribution, entropy, variational principles, spontaneous symmetry breaking, linear response theory, and phase transitions. * Know and understand the theoretical description of magnetic systems and disordered systems, with particular emphasis on the Ising model, mean-field approximations, effective Hamiltonians, and critical phenomena in finite dimensions. * Know and understand the physical properties of spin glasses, including quenched disorder, frustration, ergodicity breaking, overlap order parameters, memory and chaos effects, and the phenomenology of experimental spin-glass systems. * Know and understand the main analytical methods used in the study of disordered systems, including replica theory, cavity methods, TAP equations, replica-symmetric and replica-symmetry-breaking solutions, and the Random Energy Model. * Know and understand stochastic growth processes and nonequilibrium statistical physics models, including the Edwards–Wilkinson and Kardar–Parisi–Zhang equations, scaling properties, and the relation between KPZ dynamics, Burgers equation, and directed polymers in random media. * Know and understand the statistical physics approach to optimization and constraint satisfaction problems on sparse random graphs, including Erdős–Rényi random graphs, graph coloring, random XOR-SAT problems, and message-passing algorithms such as warning propagation, belief propagation, and survey propagation. * Apply the acquired theoretical knowledge to solve exercises involving mean-field models, phase transitions, spin-glass order parameters, stochastic processes, and scaling arguments. * Apply statistical mechanics methods and probabilistic reasoning to the analysis of complex systems characterized by disorder, frustration, and randomness. * Apply the acquired knowledge to the formulation and interpretation of algorithms for optimization problems on random graphs and constraint satisfaction problems. * Develop the ability to connect theoretical models with physical phenomena and computational applications, recognizing the interdisciplinary role of disordered systems methods in physics, mathematics, computer science, and complex systems research.

Basic probability. Equilibrium probability measures. The concept of probabilistic ensembles in statistical mechanics.

Part I – Introduction to Statistical Mechanics and Phase Transitions (6 hours) Fundamental concepts of equilibrium statistical mechanics, including the Boltzmann distribution, entropy, variational principles, and linear response theory. Magnetic systems, spontaneous symmetry breaking, the Ising model, and mean-field approaches are introduced as paradigmatic examples for the study of phase transitions and critical phenomena. Part II – Spin Glasses and Disordered Systems (20 hours) Phenomenology and theoretical description of spin glasses and systems with quenched disorder. Experimental properties, ergodicity breaking, overlap order parameters, replica methods, cavity approaches, and replica symmetry breaking are discussed within the framework of mean-field spin-glass models such as the Sherrington–Kirkpatrick model and the Random Energy Model. Part III – Stochastic Growth and KPZ Universality (10 hours) Introduction to stochastic surface growth models and nonequilibrium statistical physics. The Edwards–Wilkinson and Kardar–Parisi–Zhang equations are presented together with their scaling properties and their connections to Burgers dynamics and directed polymers in random media. Part IV – Optimization Problems on Sparse Random Graphs (24 hours) Statistical physics methods for combinatorial optimization and constraint satisfaction problems on sparse random graphs. The course covers Erdős–Rényi graphs, graph coloring, random XOR-SAT problems, and message-passing algorithms such as belief propagation and survey propagation, together with the limitations of replica-symmetric approximations.

The course comprises 54 hours of theoretical lectures and 6 hours of numerical experiments.

All lectures will be held in presence.

General Statistical Mechanics * Mehran Kardar, Statistical Physics of Particles and Statistical Physics of Fields, Cambridge University Press. Useful for the introductory part on statistical mechanics, phase transitions, scaling, and field-theory concepts. * My lecture notes for the course on Complex Systems that wil be available on the course portal. Disordered Systems and Spin Glasses * Marc Mézard, Giorgio Parisi, and Miguel Virasoro, Spin Glass Theory and Beyond, World Scientific. A classic reference on replica methods, spin glasses, and replica symmetry breaking. * Fischer, Konrad H., and John A. Hertz. Spin glasses. No. 1. Cambridge University Press, 1993. Another classic in the field. KPZ and Nonequilibrium Growth * Timothy Halpin-Healy and Yi-Cheng Zhang, “Kinetic Roughening Phenomena, Stochastic Growth, Directed Polymers and All That,” Physics Reports (1995). A standard review on KPZ universality and stochastic growth. * Ivan Corwin, “The Kardar–Parisi–Zhang Equation and Universality Class,” Random Matrices: Theory and Applications (2012). A modern introduction to KPZ theory. Optimization and Sparse Random Graphs * Marc Mézard and Andrea Montanari, Information, Physics, and Computation, Oxford University Press. A key reference for belief propagation, cavity methods, random graphs, satisfiability problems, and optimization. * F. Krzakala et al. (eds.), Statistical Physics, Optimization, Inference, and Message-Passing Algorithms, Oxford University Press (2015). Lecture notes and reviews on message-passing algorithms and constraint satisfaction problems. Additional / Advanced References * Patrick Charbonneau et al., Spin Glass Theory and Far Beyond, World Scientific World Scientific (2023). A modern and broad overview of spin glasses, optimization, machine learning, and complex systems applications. * Francesco Zamponi, Mean Field Theory of Spin Glasses (lecture notes). Useful as supplementary material for replica and cavity methods. Available at https://arxiv.org/abs/1008.4844

Dispense; Esercizi; Esercizi risolti; Strumenti di simulazione;

Modalita di esame: Prova orale obbligatoria; Elaborato progettuale individuale;

Exam: Compulsory oral exam; Individual project;

... The final examination is aimed at assessing the acquisition of the expected knowledge and skills through an oral examination and the presentation of a project, both taking place on the same day. In order to evaluate the achievement of the learning objectives, including both the understanding of the theoretical concepts and the ability to apply them, the examination is divided into two parts. The first part consists of an oral examination focused on the theoretical topics covered during the course. Students are expected to discuss and critically analyze the fundamental concepts and methods of the statistical physics of disordered systems, including phase transitions, spin glasses, stochastic growth processes, and optimization problems on random graphs. Particular attention is devoted to the ability to formulate rigorous arguments, connect different theoretical frameworks, and discuss the physical interpretation of the models presented during the course. The second part consists of the presentation and discussion of a project developed by the student on a topic related to the course. The project may involve theoretical calculations, numerical simulations, analytical studies, or the discussion of relevant scientific literature. This part of the examination is aimed at evaluating the student’s ability to independently apply the methods introduced during the course, critically analyze results, and present scientific material in a clear and coherent way. The final grade is determined by considering both parts of the examination, with equal weight assigned to the oral discussion and to the project presentation. Honors (“cum laude”) may be awarded when the student demonstrates an outstanding level of understanding, methodological rigor, critical thinking, and clarity of presentation in both parts of the examination.

Gli studenti e le studentesse con disabilita o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unita Special Needs, al fine di permettere al/la docente la declinazione piu idonea in riferimento alla specifica tipologia di esame.