Critical and Disordered Systems at ICTP-SAIFR
Statistical mechanics of disordered systems
Half-course, summer 2026 · Course description
Lecture notes
- Introduction & motivation
- Equilibrium statistical mechanics: the Curie–Weiss model
- The spherical spin glasses – annealed average
- The replica method and replica symmetry
- Stability of replica symmetry and replica symmetry breaking
- Dynamics: the cavity method and equilibrium
- Dynamics: aging
- Complexity of metastable states
- Franz–Parisi potential
Exercises
- Mathematical prerequisites
- Equilibrium statistical mechanics
- Annealed averages & Langevin equations
- Replica symmetry for Sherrington–Kirkpatrick
- Replica symmetry breaking for Sherrington–Kirkpatrick
- The fluctuation-dissipation theorem
- DMFT by path integration
- Spin glass theory and beyond
Random matrix theory
Half-course, spring 2025 · Course description · Syllabus
Lecture notes
- Introduction & motivation
- The GOE and its JPDF of eigenvalues
- From JPDF to spectral density
- The resolvent
- The resolvent several ways: cavity & replicas
- The resolvent several ways: supersymmetry
- The resolvent several ways: dynamics
- Free probability
- Localization
- Sparse matrices
- Isolated eigenvalues and low-rank perturbations
- Asymmetric random matrices
- Edge statistics
Homework assignments
- Problem 1.6 Random matrix theory in Jim Sethna’s textbook
- Dyson Brownian motion & Jacobian for Hermitian random matrices
- Coulomb gas for Wishart
- Cavity method & replicas for Wishart
- Factorization of free products & R transform for Wishart
- Population annealing on random regular graphs